Isometry groups of non-positively curved spaces: discrete subgroups
We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour…
Authors: P.-E. Caprace, N. Monod
ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS PIERRE-EMMANUEL CAPRA CE* AND NICOLAS MONOD ‡ Abstra t. W e study latties in non-p ositiv ely urv ed metri spaes. Borel densit y is established in that setting as w ell as a form of Mosto w rigidit y . A on v erse to the at torus theorem is pro vided. Geometri arithmetiit y results are obtained after a detour through sup errigidit y and arithmetiit y of abstrat latties. Residual niteness of latties is also studied. Riemannian symmetri spaes are haraterised amongst CA T(0) spaes admitting latties in terms of the existene of parab oli isometries. 1. Intr odution Latties in semi-simple algebrai groups ha v e a tan talisingly ri h struture; they inlude arithmeti groups and more generally S-arithmeti groups o v er arbitrary harateristis. The nature of these groups is shap ed in part b y the fat that they are realised as isometries of a anonial non-p ositiv ely urv ed spae: the asso iated Riemannian symmetri spae, or BruhatTits building, or a pro dut of b oth t yp es. Man y other groups of rather div erse origins share this prop ert y to o ur as latties in non-p ositiv ely urv ed spaes, singular or not: The fundamen tal group of a losed Riemannian manifold of non-p ositiv e setionnal urv ature. Here the spae ated up on is the univ ersal o v ering, whi h is a Hadamard manifold. Man y Gromo v-h yp erb oli groups admit a prop erly dison tin uous o ompat ation on some CA T( − 1 ) spae b y isometries. Amongst the examples arising in this w a y are h yp erb oli Co xeter groups [ Mou88 ℄, C ′ ( 1 6 ) and C ′ ( 1 4 ) - T (4) small anellation groups [ Wis04 ℄, 2 -dimensional 7 -systoli groups [ J06 ℄. It is in fat a w ell kno wn op en problem of M. Gromo v to onstrut an example of a Gromo v-h yp erb oli group whi h is not a CA T(0) group (see [ Gro93 , 7.B℄; also Remark 2.3(2) in Chapter I I I. Γ of [ BH99 ℄). In [ BM00b ℄, striking examples of nitely presen ted simple groups are onstruted as latties in a pro dut of t w o lo ally nite trees. T ree latties w ere previously studied in [ BL01 ℄. A minimal adjoin t KaMo o dy group o v er a nite eld, as dened b y J. Tits [ Tit87 ℄, is endo w ed with t w o B N -pairs whi h yield strongly transitiv e ations on a pair of t winned buildings. When the order of the ground eld is large enough, the Ka Mo o dy group is a lattie in the pro dut of these t w o buildings [ Rém99 ℄. Subsuming all the ab o v e examples, w e dene a CA T(0) lattie as a pair (Γ , X ) onsisting of a prop er CA T(0) spae X with o ompat isometry group Is( X ) and a lattie subgroup Key wor ds and phr ases. Lattie, arithmeti group, non-p ositiv e urv ature, CA T(0) spae, lo ally ompat group. *F.N.R.S. Resear h Asso iate. ‡ Supp orted in part b y the Swiss National Siene F oundation. 1 2 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Γ < Is( X ) , i.e. a disrete subgroup of nite in v arian t o v olume (the ompat-op en top ology mak es Is( X ) a lo ally ompat seond oun table group whi h is th us anonially endo w ed with Haar measures). W e sa y that (Γ , X ) is uniform if Γ is o ompat in Is( X ) or, equiv a- len tly , if the quotien t Γ \ X is ompat; that ase orresp onds to Γ b eing a CA T(0) group in the usual terminology . Amongst CA T(0) latties, the most imp ortan t, and also the b est understo o d, notably through the w ork of G. Margulis, onsist undoubtedly of those arising from latties in semi- simple groups o v er lo al elds. It is therefore natural to address t w o sets of questions. (a) What pr op erties of these latti es ar e shar e d by al l CA T(0) latti es? (b) What pr op erties har aterise them within the lass of CA T(0) latti es? This artile is dev oted to the study of CA T(0) latties and en tres largely around the ab o v e questions, though w e also address the general question of the in terpla y b et w een the algebrai struture of a CA T(0) lattie and the geometri prop erties of the underlying spae. Some of the te hniques established in the presen t pap er ha v e b een used in a subsequen t in v estigation of latties in pro duts of KaMo o dy groups [ CM08d ℄. W e shall no w desrib e the main results of this artile; for man y of them, the ore of the text will on tain a stronger, more preise but p erhaps more p onderous v ersion. Our notation is standard, as realled in the Notation setion of the ompanion pap er [ CM08 ℄. W e refer to the latter for terminology and shall quote it freely . . Geometri Borel densit y . As a link b et w een the general theory exp osed in [ CM08 ℄ and the study of CA T(0) latties, w e prop ose the follo wing analogue of A. Borel's densit y theorem [ Bor60 ℄. Theorem 1.1. L et X b e a pr op er CA T(0) sp a e, G a lo al ly omp at gr oup ating ontin- uously by isometries on X and Γ < G a latti e. Supp ose that X has no Eulide an fator. If G ats minimal ly without xe d p oint at innity, so do es Γ . This onlusion fails for spaes with a Eulidean fator. The theorem will b e established more generally for losed subgroups with nite in v arian t o v olume. It should b e ompared to (and an of ourse b e gainfully om bined with) a similar densit y prop ert y of normal subgroups established as Theorem 1.10 in [ CM08 ℄. Remark 1.2. Theorem 1.1 applies to general prop er CA T(0) spaes. It implies in partiular the lassial Borel densit y theorem (see the end of Setion 2 ). As with lassial Borel densit y , w e shall use this densit y statemen t to deriv e statemen ts ab out the en traliser, normaliser and radial of latties in Setion 2 . A more elemen tary v arian t of the ab o v e theorem sho ws that a large lass of groups ha v e rather restrited ations on prop er CA T(0) spaes; as an appliation, one sho ws: A ny isometri ation of R. Thompson 's gr oup F on any pr op er CA T(0) sp a e X has a xe d p oint in X , see Corollary 2.3 . Theorem 1.1 also pro vides additional information ab out the totally dis- onneted groups D j o urring in Theorem 1.6 in [ CM08 ℄. . ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 3 Latties: Eulidean fator, b oundary , irreduibilit y and Mosto w rigidit y . Reall that the Flat T orus the or em , originating in the w ork of GromollW olf [ GW71 ℄ and La wson Y au [ L Y72 ℄, asso iates Eulidean subspaes R n to an y subgroup Z n of a CA T(0) group, see [ BH99 , I I.7℄. (In the lassial setting, when the CA T(0) group is giv en b y a ompat non-p ositiv ely urv ed manifold, this amoun ts to the seemingly more symmetri statemen t that su h a subgroup exists if and only if there is a at torus is the manifold.) The on v erse is a w ell kno wn op en problem stated b y M. Gromo v in [ Gro93 , 6 . B 3 ℄; for manifolds see S.-T. Y au, problem 65 in [ Y au82 ℄). P oin t ( i ) in the follo wing result is a (v ery partial) answ er; in the sp eial ase of o ompat Riemannian manifolds, this w as the main result of P . Eb erlein's artile [ Eb e83 ℄. Theorem 1.3. L et X b e a pr op er CA T(0) sp a e, G < Is( X ) a lose d sub gr oup ating minimal ly and o omp atly on X and Γ < G a nitely gener ate d latti e. Then: (i) If the Eulide an fator of X has dimension n , then Γ p ossesses a nite index sub- gr oup Γ 0 whih splits as Γ 0 ≃ Z n × Γ ′ . Mor e over, the dimension of the Eulide an fator is har aterise d as the maximal r ank of a fr e e A b elian normal sub gr oup of Γ . (ii) G has no xe d p oint at innity; the set of Γ -xe d p oints at innity is ontaine d in the (p ossibly empty) b oundary of the Eulide an fator. P oin t ( ii ) is partiularly useful in onjution with the man y results assuming the absene of xed p oin ts at innit y in [ CM08 ℄. In addition, it is already a rst indiation that the mere existene of a (nitely generated) lattie is a serious restrition on a prop er CA T(0) spae ev en within the lass of o ompat minimal spaes. W e reall that E. Hein tze [ Hei74 ℄ pro dued simply onneted negativ ely urv ed Riemannian manifolds that are homogeneous (in partiular, o ompat) but ha v e a p oin t at innit y xed b y all isometries. Sine a CA T(0) lattie onsists of a group and a spae, there are t w o natural notions of irreduibilit y: of the group or of the spae. In the ase of latties in semi-simple groups, the t w o notions are kno wn to oinide b y a result of Margulis [ Mar91 , I I.6.7℄. W e pro v e that this is the ase for CA T(0) latties as ab o v e. Theorem 1.4. In the setting of The or em 1.3 , Γ is irr e duible as an abstr at gr oup if and only if for any nite index sub gr oup Γ 1 and any Γ 1 -e quivariant de omp osition X = X 1 × X 2 with X i non- omp at, the pr oje tion of Γ 1 to b oth Is( X i ) is non-disr ete. The om bination of Theorem 1.4 , Theorem 1.3 and of an appropriate form of sup errigidit y allo w us to giv e a CA T(0) v ersion of Mosto w rigidit y for reduible spaes (Setion 4.E ). . Geometri arithmetiit y . W e no w exp ose results giving p erhaps unexp etedly strong onlusions for CA T(0) latties b oth for the group and for the spae. These results w ere announed in [ CM08e ℄ in the ase of CA T(0) groups; the presen t setting of nitely generated latties is more general sine CA T(0) groups are nitely generated ( f. Lemma 3.3 b elo w). W e reall that an isometry g is parab oli if the translation length inf x ∈ X d ( g x, x ) is not a hiev ed. F or general CA T(0) spaes, parab oli isometries are not w ell understo o d; in fat, ruling out their existene an sometimes b e the essen tial diult y in rigidit y statemen ts. Theorem 1.5. L et (Γ , X ) b e an irr e duible nitely gener ate d CA T(0) latti e with X ge o desi- al ly omplete. Assume that X p ossesses some p ar ab oli isometry. 4 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD If Γ is r esidual ly nite, then X is a pr o dut of symmetri sp a es and BruhatTits build- ings. In p artiular, Γ is an arithmeti latti e unless X is a r e al or omplex hyp erb oli sp a e. If Γ is not r esidual ly nite, then X stil l splits o a symmetri sp a e fator. Mor e over, the nite r esidual Γ D of Γ is innitely gener ate d and Γ / Γ D is an arithmeti gr oup. (Reall that the nite residual of a group is the in tersetion of all nite index subgroups.) W e single out a purely geometri onsequene. Corollary 1.6. L et (Γ , X ) b e a nitely gener ate d CA T(0) latti e with X ge o desi al ly om- plete. Then X p ossesses a p ar ab oli isometry if and only if X ∼ = M × X ′ , wher e M is a symmetri sp a e of non- omp at typ e. Without the assumption of geo desi ompleteness, w e still obtain an arithmetiit y state- men t when the underlying spae admits some parab oli isometry that is neutral , i.e. whose displaemen t length v anishes. Neutral parab oli isometries are ev en less understo o d, not ev en for their dynamial prop erties (whi h an b e ompletely wild at least in Hilb ert spae [ Ede64 ℄); as for familiar examples, they are pro vided b y unip oten t elemen ts in semi- simple algebrai groups. Theorem 1.7. L et (Γ , X ) b e an irr e duible nitely gener ate d CA T(0) latti e. If X admits any neutr al p ar ab oli isometry, then either: (i) Is( X ) is a r ank one simple Lie gr oup with trivial entr e; or: (ii) Γ has a normal sub gr oup Γ D suh that Γ / Γ D is an arithmeti gr oup. Mor e over, Γ D is either nite or innitely gener ate d. W e turn to another t yp e of statemen t of arithmetiit y/geometri sup errigidit y . Ha ving established an abstrat arithmetiit y theorem (presen ted b elo w as Theorem 1.9 ), w e an app eal to our geometri results and pro v e the follo wing. Theorem 1.8. L et (Γ , X ) b e an irr e duible nitely gener ate d CA T(0) latti e with X ge o desi- al ly omplete. Assume that Γ p ossesses some faithful nite-dimensional line ar r epr esenta- tion ( in har ateristi 6 = 2 , 3) . If X is r e duible, then Γ is an arithmeti latti e and X is a pr o dut of symmetri sp a es and BruhatTits buildings. Setion 6 on tains more results of this nature but also demonstrates b y a family of exam- ples that some of the in triaies in the more detailed statemen ts reet indeed the existene of more exoti pairs (Γ , X ) . . Abstrat arithmetiit y . When preparing for the pro of of our geometri arithmetiit y statemen ts, w e are led to study irreduible latties in pro duts of general top ologial groups in the abstrat. Building notably on ideas of Margulis, w e stablish the follo wing arithmeti- it y statemen t (for whi h w e reall that the quasi-en tre Q Z of a top ologial group is the subset of elemen ts with op en en traliser). Theorem 1.9. L et Γ < G = G 1 × · · · × G n b e an irr e duible nitely gener ate d latti e, wher e e ah G i is any lo al ly omp at gr oup. If Γ admits a faithful Zariski-dense r epr esentation in a semi-simple gr oup over some eld of har ateristi 6 = 2 , 3 , then the amenable r adi al R of G is omp at and the quasi- entr e Q Z ( G ) is virtual ly ontaine d in Γ · R . F urthermor e, up on r eplaing G by a nite index ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 5 sub gr oup, the quotient G/R splits as G + × Q Z ( G/R ) wher e G + is a semi-simple algebr ai gr oup and the image of Γ in G + is an arithmeti latti e. In partiular, the quasi-en tre Q Z ( G/R ) is disr ete . In shorter terms, this theorem states that up to a ompat extension, G is the diret pro dut of a semi-simple algebrai group b y a (p ossibly trivial) disrete group, and that the image of Γ in the non-disrete part is an arithmeti group. The assumption on the harateristi an b e sligh tly w eak ened. In the ourse of the pro of, w e haraterise all irreduible nitely generated latties in pro duts of the form G = S × D where S is a semi-simple Lie group and D a totally disonneted group (Theorem 5.18 ). In partiular, it turns our that D m ust neessarily b e lo ally pronite b y analyti. The orresp onding question for simple algebrai groups instead of Lie groups is also in v estigated (Theorem 5.20 ). . Unique geo desi extension. Complete simply onneted Riemannian manifolds of non- p ositiv e urv ature, sometimes also alled Hadamard manifolds, form a lassial family of prop er CA T(0) spaes to whi h the preeding results ma y b e applied. In fat, the natural lass to onsider in our on text onsists of those prop er CA T(0) spaes in whi h ev ery geo- desi segmen t extends uniquely to a bi-innite geo desi line. Clearly , this lass on tains all Hadamard manifolds, but it presumably on tains more examples. It is, ho w ev er, somewhat restrited with resp et to the main thrust of the presen t w ork sine it do es not allo w for, sa y , simpliial omplexes; aordingly , the onlusions of the theorem b elo w are also more stringen t. Theorem 1.10. L et X b e a pr op er CA T(0) sp a e with uniquely extensible ge o desis. Assume that Is( X ) ats o omp atly without xe d p oints at innity. (i) If X is irr e duible, then either X is a symmetri sp a e or Is( X ) is disr ete. (ii) If Is( X ) p ossesses a nitely gener ate d non-uniform latti e Γ whih is irr e duible as an abstr at gr oup, then X is a symmetri sp a e (without Eulide an fator). (iii) Supp ose that Is( X ) p ossesses a nitely gener ate d latti e Γ (if Γ is uniform, this is e quivalent to the ondition that Γ is a disr ete o omp at gr oup of isometries of X ). If Γ is irr e duible (as an abstr at gr oup) and X is r e duible, then X is a symmetri sp a e (without Eulide an fator). In the sp eial ase of Hadamard manifolds, statemen t (i) w as kno wn under the assumption that Is( X ) satises the dualit y ondition (without assuming that Is( X ) ats o ompatly without xed p oin ts at innit y). This is due to P . Eb erlein (Prop osition 4.8 in [ Eb e82 ℄). Lik ewise, statemen t (iii) for manifolds is Prop osition 4.5 in [ Eb e82 ℄. More reen tly , F arbW ein b erger [ FW06 ℄ in v estigated analogous questions for aspherial manifolds. . Latties and the de Rham deomp osition. In [ CM08 ℄, w e pro v ed a de Rham deomp osition (1.i) X ′ ∼ = X 1 × · · · × X p × R n × Y 1 × · · · × Y q for prop er CA T(0) spaes X with nite-dimensional Tits b oundary and su h that Is( X ) has no xed p oin t at innit y , see A ddendum 1.8 in [ CM08 ℄. (Here X ′ ⊆ X is the anonial 6 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD minimal in v arian t subspae, and w e reall that X ′ = X e.g. when X is geo desially omplete and admits a o ompat lattie b y Lemma 3.13 in [ CM08 ℄.) It turns out that this de Rham deomp osition is an in v arian t of CA T(0) groups in the follo wing sense (see Corollary 4.14 ). Theorem 1.11. L et X b e a pr op er CA T(0) sp a e and Γ < Is( X ) b e a gr oup ating pr op erly dis ontinuously and o omp atly. Then any other suh sp a e admitting a pr op er o omp at Γ -ation has the same numb er of fators in ( 1.i ) and the Eulide an fator has same dimension. ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 7 Contents 1. In tro dution 1 2. An analogue of Borel densit y 8 2.A. Fixed p oin ts at innit y 8 2.B. Geometri densit y for subgroups of nite o v olume 9 2.C. The limit set of subgroups of nite o v olume 11 3. CA T(0) latties, I: the Eulidean fator 12 3.A. Preliminaries on latties 12 3.B. V ariations on Auslander's theorem 13 3.C. Latties, the Eulidean fator and xed p oin ts at innit y 14 4. CA T(0) latties, I I: pro duts 18 4.A. Irreduible latties in CA T(0) spaes 18 4.B. The h ull of a lattie 20 4.C. On the anonial disrete k ernel 21 4.D. Residually nite latties 22 4.E. Strong rigidit y for pro dut spaes 23 5. Arithmetiit y of abstrat latties 24 5.A. Sup errigid pairs 26 5.B. Boundary maps 29 5.C. Radial sup errigidit y 30 5.D. Latties with non-disrete ommensurators 31 5.E. Latties in pro duts of Lie and totally disonneted groups 32 5.F. Latties in general pro duts 34 6. Geometri arithmetiit y 37 6.A. CA T(0) latties and parab oli isometries 37 6.B. Arithmetiit y of linear CA T(0) latties 39 6.C. A family of examples 40 7. A few questions 42 Referenes 44 8 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD 2. An analogue of Borel density Before disussing our analogue of Borel's densit y theorem [ Bor60 ℄ in Setion 2.B b elo w, w e presen t a more elemen tary phenomenon based on o-amenabilit y . 2.A. Fixed p oin ts at innit y. Reall that a subgroup H of a top ologial group G is o-amenable if an y on tin uous ane G -ation on a on v ex ompat set (in a Hausdor lo ally on v ex top ologial v etor spae) has a xed p oin t whenev er it has an H -xed p oin t. The argumen ts of A damsBallmann [ AB98 ℄ imply the follo wing preliminary step to w ards Theorem 2.4 : Prop osition 2.1. L et G b e a top olo gi al gr oup with a ontinuous isometri ation on a pr op er CA T(0) sp a e X without Eulide an fator. Assume that the G -ation is minimal and do es not have a glob al xe d p oint in ∂ X . Then any o-amenable sub gr oup of G stil l has no glob al xe d p oint in ∂ X . Pr o of. Supp ose for a on tradition that a o-amenable subgroup H < G xes ξ ∈ ∂ X . Then G preserv es a probabilit y measure µ on ∂ X and w e obtain a on v ex funtion f : X → R b y in tegrating Busemann funtions against this measure; as in [ AB98 ℄, the o yle equation for Busemann funtions (see 2 in [ CM08 ℄) imply that f is G -in v arian t up to onstan ts. The argumen ts therein sho w that f is onstan t and that µ is supp orted on at p oin ts. Ho w ev er, in the absene of a Eulidean fator, the set of at p oin ts has a unique irumen tre when non-empt y [ AB98 , 1.7℄; this pro vides a G -xed p oin t, a on tradition. Com bining the ab o v e with the splitting metho ds used in Theorem 4.3 in [ CM08 ℄, w e reord a onsequene sho wing that the exat onlusions of the A damsBallmann theo- rem [ AB98 ℄ hold under m u h w eak er assumptions than the amenabilit y of G . Corollary 2.2. L et G b e a top olo gi al gr oup with a ontinuous isometri ation on a pr op er CA T(0) sp a e X . Assume that G ontains two ommuting o-amenable sub gr oups. Then either G xes a p oint at innity or it pr eserves a Eulide an subsp a e in X . W e emphasise that one an easily onstrut a w ealth of examples of highly non-amenable groups satisfying these assumptions. F or instane, giv en any group Q , the restrited wreath pro dut G = Z ⋉ L n ∈ Z Q on tains the pair of omm uting o-amenable groups H + = L n ≥ 0 Q and H − = L n< 0 Q , see [ MP03 ℄. (In fat, one an ev en arrange for H ± to b e onjugated up on replaing Z b y the innite dihedral group.) F or similar reasons, w e dedue the follo wing xed-p oin t prop ert y for R. Thompson's group F := g i , i ∈ N | g − 1 i g j g i = g j +1 ∀ j > i ; this xed-p oin t result explains wh y the strategy prop osed in [ F ar08 ℄ to dispro v e amenabilit y of F with the A damsBallmann theorem annot w ork. Corollary 2.3. A ny F -ation by isometries on any pr op er CA T(0) sp a e X has a xe d p oint in X . Pr o of of Cor ol lary 2.2 . W e assume that G has no xed p oin t at innit y . By Prop osition 4.1 in [ CM08 ℄, there is a minimal non-empt y losed on v ex G -in v arian t subspae. Up on on- sidering the Eulidean deomp osition [ BH99 , I I.6.15℄ of the latter, w e an assume that X is G -minimal and without Eulidean fator and need to sho w that G xes a p oin t in X . Let H ± < G b e the omm uting o-amenable groups. In view of Prop osition 2.1 , b oth at without xed p oin t at innit y . In partiular, w e ha v e an ation of H = H + × H − without xed p oin t at innit y and the splitting theorem from [ Mon06 ℄ pro vides us with a ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 9 anonial subspae X + × X − ⊆ X with omp onen t-wise and minimal H -ation. All of ∂ X + is xed b y H − , whi h means that this b oundary is empt y . Sine X is prop er, it follo ws that X + is b ounded and hene redued to a p oin t b y minimalit y . Th us H + xes a p oin t in X ⊆ X and o-amenabilit y implies that G xes a probabilit y measure µ on X . If µ w ere supp orted on ∂ X , the pro of of Prop osition 2.1 w ould pro vide a G -xed p oin t at innit y , whi h is absurd. Therefore µ ( X ) > 0 . No w ho ose a b ounded set B ⊆ X large enough so that µ ( B ) > µ ( X ) / 2 . Then an y G -translate of B m ust meet B . It follo ws that G has a b ounded orbit and hene a xed p oin t as laimed. Pr o of of Cor ol lary 2.3 . W e refer to [ CFP96 ℄ for a detailed in tro dution to the group F . In partiular, F an b e realised as the group of all orien tation-preserving pieewise ane homeomorphisms of the in terv al [0 , 1] that ha v e dy adi breakp oin ts and slop es 2 n with n ∈ Z . Giv en a subset A ⊆ [0 , 1] w e denote b y F A < F the subgroup supp orted on A . W e laim that whenev er A has non-empt y in terior, F A is o-amenable in F . The argumen t is analogous to [ MP03 ℄ and to [ GM07 , 4.F℄; indeed, in view of the alternativ e denition of F just realled, one an ho ose a sequene { g n } in F su h that g n A on tains [1 /n, 1 − 1 /n ] and th us F g n A on tains F [1 /n, 1 − 1 /n ] . Consider the ompat spae of means on F /F A , namely nitely additiv e measures, endo w ed with the w eak-* top ology from the dual of ℓ ∞ ( F /F A ) . An y aum ulation p oin t µ of the sequene of Dira masses at g − 1 n F A will b e in v arian t under the union F ′ of the groups F [1 /n, 1 − 1 /n ] . No w F ′ is the k ernel of the deriv ativ e homomorphism F → 2 Z × 2 Z at the pair of p oin ts { 0 , 1 } . In partiular, F ′ is o-amenable in F and th us the F ′ -in v ariane of µ implies that there is also a F -in v arian t mean on F /F A , whi h is one of the haraterisations of o-amenabilit y [ Eym72 ℄. Let no w X b e an y prop er CA T(0) spae with an F -ation b y isometries. W e an assume that F has no xed p oin t at innit y and therefore w e an also assume that X is minimal b y Prop osition 4.1 in [ CM08 ℄. The ab o v e laim pro vides us with man y pairs of omm uting o-amenable subgroups up on taking disjoin t sets of non-empt y in terior. Therefore, Corol- lary 2.2 sho ws that X ∼ = R n for some n . In partiular the isometry group is linear. Sine F is nitely generated (b y g 0 and g 1 in the ab o v e presen tation, ompare also [ CFP96 ℄), Malev's theorem [ Mal40 ℄ implies that the image of F is residually nite. The deriv ed subgroup of F (whi h iniden tally oinides with the group F ′ in tro dued ab o v e) b eing simple [ CFP96 ℄, it follo ws that it ats trivially . It remains only to observ e that t w o omm uting isometries of R n alw a ys ha v e a ommon xed p oin t in R n , whi h is a matter of linear algebra. The ab o v e reasoning an b e adapted to yield similar results for bran h groups and related groups; w e shall address these questions elsewhere. 2.B. Geometri densit y for subgroups of nite o v olume. The follo wing geometri densit y theorem generalises Borel's densit y (see Prop osition 2.8 b elo w) and on tains Theo- rem 1.1 from the In tro dution. Theorem 2.4. L et G b e a lo al ly omp at gr oup with a ontinuous isometri ation on a pr op er CA T(0) sp a e X without Eulide an fator. If G ats minimal ly and without glob al xe d p oint in ∂ X , then any lose d sub gr oup with nite invariant ovolume in G stil l has these pr op erties. Remark 2.5. F or a related statemen t without the assumption on the Eulidean fator of X or on xed p oin ts at innit y , see Theorem 3.14 b elo w. Pr o of. Retain the notation of the theorem and let Γ < G b e a losed subgroup of nite in v arian t o v olume. In partiular, Γ is o-amenable and th us has no xed p oin ts at innit y 10 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD b y Prop osition 2.1 . By Prop osition 4.1 in [ CM08 ℄, there is a minimal non-empt y losed on v ex Γ -in v arian t subset Y ⊆ X and it remains to sho w Y = X . Cho ose a p oin t x 0 ∈ X and dene f : X → R b y f ( x ) = Z G/ Γ d ( x, g Y ) − d ( x 0 , g Y ) dg . This in tegral on v erges b eause the in tegrand is b ounded b y d ( x, x 0 ) . The funtion f is on tin uous, on v ex (b y [ BH99 , I I.2.5(1)℄) and quasi-in v arian t in the sense that it satises (2.i) f ( hx ) = f ( x ) − f ( hx 0 ) ∀ h ∈ G. Sine G ats minimally and without xed p oin t at innit y , this implies that f is onstan t (see Setion 2 in [ AB98 ℄; alternativ ely , when ∂ X is nite-dimensional, it follo ws from The- orem 1.10 in [ CM08 ℄ sine ( 2.i ) implies that f is in v arian t under the deriv ed subgroup G ′ ). In partiular, d ( x, g Y ) is ane for all g . It follo ws that for all x ∈ X the losed set Y x = z ∈ X : d ( z , Y ) = d ( x, Y ) is on v ex. W e laim that it is parallel to Y in the sense that d ( z , Y ) = d ( y , Y x ) for all z ∈ Y x and all y ∈ Y . Indeed, on the one hand d ( z , Y ) is onstan t o v er z ∈ Y x b y denition, and on the other hand d ( y , Y x ) is onstan t b y minimalit y of Y sine d ( · , Y x ) is a on v ex Γ -in v arian t funtion. In partiular, Y x is Γ -equiv arian tly isometri to Y via nearest p oin t pro jetion (ompare [ BH99 , I I.2.12℄) and ea h Y x is Γ -minimal. A t this p oin t, Remarks 39 in [ Mon06 ℄ sho w that there is an isometri Γ -in v arian t splitting X ∼ = Y × T . It remains to sho w that the spae of omp onen ts T is redued to a p oin t. Let th us s, t ∈ T and let m b e their midp oin t. Applying the ab o v e reasoning to the hoie of minimal set Y 0 orresp onding to Y × { m } , w e dedue again that the distane to Y 0 is an ane funtion on X . Ho w ev er, this funtion is preisely the distane funtion d ( · , m ) in T omp osed with the pro jetion X → T . Being non-negativ e and ane on [ s, t ] , it v anishes on that segmen t and hene s = t . Remark 2.6. When Γ is o ompat in G , the pro of an b e shortened b y in tegrating just d ( x, g Y ) in the denition of f ab o v e. Corollary 2.7. L et X b e a pr op er CA T(0) sp a e without Eulide an fator suh that G = Is( X ) ats minimal ly without xe d p oint at innity, and let Γ < G b e a lose d sub gr oup with nite invariant ovolume. Then: (i) Γ has trivial amenable r adi al. (ii) The entr aliser Z G (Γ) is trivial. (iii) If Γ is nitely gener ate d, then is has nite index in its normaliser N G (Γ) and the latter is a nitely gener ate d latti e in G . Pr o of. (i) and (ii) follo w b y the same argumen t as in the pro of of Theorem 1.10 in [ CM08 ℄. F or (iii) w e follo w [ Mar91 , Lemma I I.6.3℄. Sine Γ is losed and oun table, it is disrete b y Baire's ategory theorem and th us is a lattie in G . Sine it is nitely generated, its automorphism group is oun table. By (ii), the normaliser N G (Γ) maps injetiv ely to Aut(Γ) and hene is oun table as w ell. Th us N G (Γ) , b eing losed in G , is disrete b y applying Baire again. Sine it on tains the lattie Γ , it is itself a lattie and the index of Γ in N G (Γ) is nite. Th us N G (Γ) is nitely generated. ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 11 As p oin ted out b y P . de la Harp e, p oin t( ii ) implies in partiular that an y lattie in G is ICC (whi h means b y denition that all its non-trivial onjugay lasses are innite). As is w ell kno wn, this is the riterion ensuring that the t yp e I I 1 v on Neumann algebra asso iated to the lattie is a fator [ T ak02 , V.7℄. Finally , w e indiate wh y Theorem 2.4 implies the lassial Borel densit y theorem of [ Bor60 ℄. It sues to justify the follo wing: Prop osition 2.8. L et k by a lo al eld (A r hime de an or not), G a semi-simple k -gr oup without k -anisotr opi fators, X the symmetri sp a e or BruhatTits building asso iate d to G = G ( k ) and L < G any sub gr oup. If the L -ation on X is minimal without xe d p oint at innity, then L is Zariski-dense. Pr o of. Let ¯ L b e the ( k -p oin ts of the) Zariski losure of L . Then ¯ L is semi-simple; this follo ws e.g. from a v ery sp eial ase of Corollary 5.8 in [ CM08 ℄, whi h guaran tees that the radial of ¯ L is trivial. In the Ar himedean ase, w e ma y app eal to Karp elevi hMosto w theorem (see [ Kar53 ℄ or [ Mos55 ℄): an y semi-simple subgroup has a totally geo desi orbit in the symmetri spae. So the only semi-simple subgroup ating minimally is G itself. In the non-Ar himedean ase, w e ould app eal to E. Landv ogt funtorialit y theorem [ Lan00 ℄ whi h w ould nish the pro of. Ho w ev er, there is an alternativ e diret and elemen tary argu- men t whi h a v oids app ealing to lo . it. and go es as follo ws. First notie that, b y the same argumen t as in the pro of of Theorem 7.4 in [ CM08 ℄ p oin t ( iv ), the k -rank of a semi-simple subgroup ating minimally equals the k -rank of G (this holds in all ases, not only in the non-Ar himedean one). Therefore, the inlusion of spherial buildings B ¯ L → B G pro vided b y the group inlusion ¯ L → G has the prop ert y that B ¯ L is a top-dimensional sub-building of B G . An elemen tary argumen t (see [ KL06 , Lemma 3.3℄) sho ws that the union Y of all apartmen ts of X b ounded b y a sphere in B ¯ L is a losed on v ex subset of X . Clearly Y is ¯ L -in v arian t, hene Y = X b y minimalit y . Therefore B ¯ L = B G , whi h nally implies that ¯ L = G . 2.C. The limit set of subgroups of nite o v olume. Let X b e a omplete CA T(0) spae and G a group ating b y isometries on X . Reall that the limit set Λ G of G is the in tersetion of the b oundary ∂ X with the losure of the orbit G.x 0 in X = X ⊔ ∂ X of an y x 0 ∈ X , this set b eing indep enden t of x 0 . Prop osition 2.9. L et G b e a lo al ly omp at gr oup ating ontinuously by isometries on a omplete CA T(0) sp a e X . If Γ < G is any lose d sub gr oup with nite invariant ovolume, then ΛΓ = Λ G . Consider the follo wing immediate orollary , whi h in the sp eial ase of Hadamard man- ifolds follo ws from the duality ondition , see 1.9.16 and 1.9.32 in [ Eb e96 ℄. Corollary 2.10. L et G b e a lo al ly omp at gr oup with a ontinuous ation by isometries on a pr op er CA T(0) sp a e. If the G -ation is o omp at, then any latti e in G has ful l limit set in ∂ X . Pr o of of Pr op osition 2.9 . W e observ e that for an y non-empt y op en set U ⊆ G there is a ompat set C ⊆ G su h that U − 1 Γ C = G . Indeed, (using an idea of Selb erg, ompare Lemma 1.4 in [ Bor60 ℄), it sues to tak e C so large that µ Γ C ) > µ (Γ \ G ) − µ (Γ U ) , 12 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD where µ denotes an in v arian t measure on Γ \ G ; an y righ t translate of Γ U in Γ \ G will then meet Γ C . No w let ξ ∈ Λ G and x 0 ∈ X . F or an y neigh b ourho o d V of ξ in ∂ X , w e shall onstrut an elemen t in ΛΓ ∩ V . Let U ⊆ G b e a ompat neigh b ourho o d of the iden tit y in G su h that U ξ ⊆ V and let { g n } b e a sequene of elemen ts of G with g n x 0 on v erging to ξ (one uses nets if X is not separable). In view of the ab o v e observ ation, there are sequenes { u n } in U and { c n } in C su h that u n g n c − 1 n ∈ Γ . The p oin ts g n c − 1 n x 0 remain at b ounded distane of g n x 0 as n → ∞ , and th us on v erge to ξ . Therefore, ho osing an aum ulation p oin t u of { u n } in U , w e see that uξ is an aum ulation p oin t of { u n g n c − 1 n x 0 } , whi h is a sequene in Γ x 0 . F or future use, w e observ e a v arian t of the ab o v e reasoning yielding a more preise fat in a simpler situation: Lemma 2.11. L et G b e a lo al ly omp at gr oup with a ontinuous o omp at ation by isometries on a pr op er CA T(0) sp a e X . L et Γ < G b e a latti e and c : R + → X a ge o desi r ay suh that G xes c ( ∞ ) . Then ther e is a se quen e { γ i } in Γ suh that γ i c ( i ) r emains b ounde d over i ∈ N . Pr o of. F or the same reason as ab o v e, there is a ompat set U ⊆ G su h that G = U Γ U − 1 . Cho ose no w { g i } su h that g i c ( i ) remains b ounded and write g i = u i γ i v − 1 i with u i , v i ∈ U . W e ha v e d ( γ i c ( i ) , c (0)) = d ( g i v i c ( i ) , u i c (0)) ≤ d ( g i v i c ( i ) , g i c ( i )) + d ( g i c ( i ) , u i c (0)) ≤ d ( v i c ( i ) , c ( i )) + d ( g i c ( i ) , c (0)) + d ( u i c (0) , c (0)) . This is b ounded indep enden tly of i b eause d ( v i c ( i ) , c ( i )) ≤ d ( v i c (0) , c (0)) sine c ( ∞ ) is G -xed. W e shall also need the follo wing: Lemma 2.12. A lo al ly omp at gr oup ontaining a nitely gener ate d sub gr oup whose lo- sur e has nite ovolume is omp atly gener ate d. Pr o of. Denoting the losure of the giv en nitely generated subgroup b y Γ , w e an write G = U Γ C as in the pro of of Prop osition 2.9 with b oth U and C ompat. Sine Γ is a lo ally ompat group on taining a nitely generated dense subgroup, it is ompatly generated and the onlusion follo ws. 3. CA T(0) la tties, I: the Eulidean f a tor 3.A. Preliminaries on latties. W e b egin this setion with a few w ell kno wn basi fats ab out general latties. Prop osition 3.1. L et G b e a lo al ly omp at se ond ountable gr oup and N ✁ G b e a lose d normal sub gr oup. (i) Given a lose d o omp at sub gr oup Γ < G , the pr oje tion of Γ on G/ N is lose d if and only if Γ ∩ N is o omp at in N . (ii) Given a latti e Γ < G , the pr oje tion of Γ on G/ N is disr ete if and only if Γ ∩ N is a latti e in N . Pr o of. See Theorem 1.13 in [ Rag72 ℄. The seond w ell kno wn result is straigh tforw ard to establish: ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 13 Lemma 3.2. L et G = H × D b e a lo al ly omp at gr oup. Given a latti e Γ < G and a omp at op en sub gr oup Q < D , the sub gr oup Γ Q := Γ ∩ ( H × Q ) is a latti e in H × Q , whih is ommensur ate d by Γ . If mor e over G/ Γ is omp at, then so is ( H × Q ) / Γ Q . (As w e shall see in Lemma 5.15 b elo w, there is a form of on v erse.) Let X b e a prop er CA T(0) spae and G = Is( X ) b e its isometry group. Giv en a disrete group Γ ating prop erly and o ompatly on X , then the quotien t G \ X is ompat and the image of Γ in G is a o ompat lattie (note that the k ernel of the map Γ → Is( X ) is nite). Con v ersely , if the quotien t G \ X is ompat, then an y o ompat lattie of G is a disrete group ating prop erly and o ompatly on X . Lemma 3.3. In the ab ove setting, G is omp atly gener ate d and Γ is nitely gener ate d. Pr o of. F or la k of nding a lassial referene, w e refer to Lemma 22 in [ MMS04 ℄). 3.B. V ariations on Auslander's theorem. Lemma 3.4. L et A = R n ⋊ O ( n ) and S b e a semi-simple Lie gr oup without omp at fator. A ny latti e Γ in G = A × S has a nite index sub gr oup Γ 0 whih splits as a dir e t pr o dut Γ 0 ∼ = Γ A × Γ ′ , wher e Γ A = Γ ∩ ( A × 1) is a latti e in ( A × 1) . Pr o of. Let V = R n denote the translation subgroup of A and U denote the losure of the pro jetion of Γ to S . The subgroup U < S is losed of nite o v olume; therefore it is either disrete or it on tains a semi-simple subgroup of p ositiv e dimension b y Borel's densit y theorem (in fat one ould b e more preise using the Main Result of [ Pra77 ℄, but this is not neessary for the presen t purp oses). On the other hand, Auslander's theorem [ Rag72 , Theorem 8.24℄ ensures that the iden tit y omp onen t of the pro jetion of Γ in S × A/V is soluble, from whi h it follo ws that U has a onneted soluble normal subgroup. Th us U is disrete. Therefore, b y Prop osition 3.1 , the group Γ A = Γ ∩ ( A × 1) is a lattie in ( A × 1) . In partiular Γ A is virtually Ab elian [ Th u97 , Corollary 4.1.13℄. Sine the pro jetion of Γ to S is a lattie in S , it is nitely generated [ Rag72 , 6.18℄. Therefore Γ p ossesses a nitely generated subgroup Λ on taining Γ A and whose pro jetion to S oinides with the pro jetion of Γ . Notie that Λ is a lattie in S × A b y [ Sim96 , Theorem 23.9.3℄; therefore Λ has nite index in Γ , whi h sho ws that Γ is nitely generated. Sine Γ A is normal in Γ , the pro jetion Γ A of Γ to A normalises the lattie Γ A and is th us virtually Ab elian. Hene Γ A is a nitely generated virtually Ab elian group whi h normalises Γ A . Therefore Γ A has a nite index subgroup whi h splits as a diret pro dut of the form Γ A × C , and the preimage Γ ′ of C in Γ is a normal subgroup whi h in tersets Γ A trivially . In partiular the group Γ ′ · Γ A ∼ = Γ ′ × Γ A is a nite index normal subgroup of Γ , as desired. Lemma 3.5. L et Γ b e a gr oup ontaining a sub gr oup of the form Γ 0 ∼ = Γ 0 S × Γ 0 A , wher e Γ 0 S is isomorphi to a latti e in a semi-simple Lie gr oup with trivial entr e and no omp at fator, and Γ 0 A is amenable. If Γ ommensur ates Γ 0 , then Γ ommensur ates b oth Γ 0 S and Γ 0 A . Pr o of. Let Γ 1 ∼ = Γ 1 S × Γ 1 A b e a onjugate of Γ 0 in Γ . The pro jetion of Γ 0 ∩ Γ 1 to Γ 0 S is a nite index subgroup of Γ 0 S . By Borel densit y theorem, it m ust therefore ha v e trivial amenable radial. In partiular the pro jetion of Γ 0 ∩ Γ 1 A to Γ 0 S is trivial. Therefore the image of the pro jetion of Γ 0 ∩ Γ 1 S (resp. Γ 0 ∩ Γ 1 A ) to Γ 0 S (resp. Γ 0 A ) is of nite index. The desired assertion follo ws. Prop osition 3.6. L et A = R n ⋊ O ( n ) , S b e a semi-simple Lie gr oup with trivial entr e and no omp at fator, D b e a total ly dis onne te d lo al ly omp at gr oup and G = S × A × D . 14 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Then any nitely gener ate d latti e Γ < G has a nite index sub gr oup Γ 0 whih splits as a dir e t pr o dut Γ 0 ∼ = Γ A × Γ ′ , wher e Γ A ⊆ Γ ∩ (1 × A × D ) is a nitely gener ate d virtual ly A b elian sub gr oup whose pr oje tion to A is a latti e. Pr o of. Let Q < D b e a ompat op en subgroup. By Lemma 3.2 , the in tersetion Γ 0 = Γ ∩ ( S × A × Q ) is a lattie in S × A × Q , whi h is ommensurated b y Γ . Sine Q is ompat, the pro jetion of Γ 0 to S × A is a lattie, to whi h w e ma y apply Lemma 3.4 . Up on replaing Γ 0 b y a nite index subgroup (whi h amoun ts to replaing Q b y an op en subgroup), this yields t w o normal subgroups Γ 0 S , Γ 0 A < Γ 0 and a deomp osition Γ 0 = Γ 0 S · Γ 0 A , where Γ 0 S ∩ Γ 0 A ⊆ Q and Γ 0 A = Γ 0 ∩ (1 × A × Q ) is a nitely generated virtually Ab elian group whose pro jetion to A is a lattie. By virtue of Lemma 3.5 , w e dedue that the image of the pro jetion of Γ to A om- mensurates a lattie in A . But the ommensurator of an y lattie in A is virtually Ab elian. Therefore, up on replaing Γ b y a nite index subgroup, it follo ws that the pro jetion of Γ to A normalises the pro jetion of Γ 0 A . W e no w dene Γ A = \ γ ∈ Γ γ Γ 0 A γ − 1 . Then the pro jetion of Γ A oinides with the pro jetion of Γ 0 A sine A is Ab elian; in parti- ular it is still a lattie. F urthermore, the subgroup Γ A is normal in Γ . W e no w pro eed as in the pro of of Lemma 3.4 . Sine the pro jetion of Γ to A is nitely generated and virtually Ab elian, w e ma y th us nd in this group a virtual omplemen t to the image of the pro jetion of Γ A . Let Γ ′ b e the preimage of this omplemen t in Γ . Then, up on replaing Γ b y a nite index subgroup, the group Γ ′ is normal in Γ and Γ = Γ A · Γ ′ . Sine Γ A is normal as w ell, the omm utator [Γ A , Γ ′ ] is on tained in the in tersetion Γ A ∩ Γ ′ , whi h is trivial b y onstrution. This nally sho ws that Γ ∼ = Γ A × Γ ′ , as desired. Remark 3.7. In the setting of Prop osition 3.6 , assume that an y ompat subgroup of D normalised b y Γ is trivial. Then Γ A ⊆ 1 × A × 1 and the pro jetion of Γ to S × D is disrete. Indeed, the denition of Γ A giv en in the pro of sho ws that it is on tained in 1 × A × γ Qγ − 1 for all γ ∈ Γ and under the urren t assumptions the in tersetion T γ Qγ − 1 is trivial. The laim ab out the pro jetion to S × D follo ws from Prop osition 3.1 . 3.C. Latties, the Eulidean fator and xed p oin ts at innit y. Giv en a prop er CA T(0) spae X and a disrete group Γ ating prop erly and o ompatly , it is a w ell kno wn op en question, going ba k to M. Gromo v [ Gro93 , 6 . B 3 ℄, to determine whether the presene of an n -dimensional at in X implies the existene of a free Ab elian group of rank n in Γ . (In the manifold ase, see problem 65 on Y au's list [ Y au82 ℄.) Here w e prop ose the follo wing theorem; the sp eial ase where X/ Γ is a ompat Riemannian manifold is the main result of Eb erlein's artile [ Eb e83 ℄ (ompare also the earlier Theorem 5.2 in [ Eb e80 ℄). Theorem 3.8. L et X b e a pr op er CA T(0) sp a e suh that G = Is( X ) ats o omp atly. Supp ose that X ∼ = R n × X ′ . (i) A ny nitely gener ate d latti e Γ < G has a nite index sub gr oup Γ 0 whih splits as a dir e t pr o dut Γ 0 ∼ = Z n × Γ ′ . (ii) If mor e over X is G -minimal ( e.g. if X is ge o desi al ly omplete), then Z n ats trivial ly on X ′ and as a latti e on R n ; the pr oje tion of Γ to Is( X ′ ) is disr ete. W e reall that o ompat latties are automatially nitely generated in the ab o v e set- ting, Lemma 3.3 . The follo wing example sho ws that, without the assumption that G ats minimally , the pro jetion of Γ to Is( X ′ ) should not b e exp eted to ha v e disrete image: ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 15 Example 3.9 . Let X b e the losed submanifold of R 3 dened b y X = { ( x, y , z ) ∈ R 3 | 1 ≤ z ≤ 2 } and onsider the follo wing Riemannian metri on X : ds 2 = dx 2 + z 2 dy 2 + dz 2 . One readily v eries that it is non-p ositiv ely urv ed; th us X is a CA T(0) manifold. Clearly X splits o a one-dimensional Eulidean fator along the x -axis. Moreo v er the group H ∼ = R 2 of all translations along the xy -plane preserv es X and ats o ompatly . Let Γ b e the subgroup of H generated b y a and b , where a : ( x, y , z ) 7→ ( x, y , z ) + ( √ 2 , 1 , 0) and b : ( x, y , z ) 7→ ( x, y , z ) + (1 , √ 2 , 0) . Then Γ ∼ = Z 2 is a o ompat lattie in Is( X ) , but no non-trivial subgroup of Γ ats trivially on the y z -fator of X . The pro jetion of Γ to the isometry group of that fator is not disrete (see Prop osition 3.1 (ii)). The ab o v e result is the on v erse to the Flat T orus Theorem when it is stated as in [ BH99 , I I.7.1℄. In partiular w e dedue that the dimension of the Eulidean de Rham fator is an in v arian t of Γ . In the manifold ase, again, this is the main p oin t of [ Eb e83 ℄. Corollary 3.10. L et X b e a pr op er CA T(0) sp a e suh that G = Is( X ) ats o omp atly and minimal ly. L et Γ < G b e a nitely gener ate d latti e. Then the dimension of the Eulide an fator of X e quals the maximal r ank of a fr e e A b elian normal sub gr oup of Γ . In order to apply Theorem 1.6 in [ CM08 ℄ and A ddendum 1.8 in [ CM08 ℄ to w ards The- orem 3.8 , w e will need the follo wing. Theorem 3.11. L et X b e a pr op er CA T(0) sp a e suh that G = Is( X ) ats o omp atly and ontains a nitely gener ate d latti e. Then X ontains a anoni al lose d onvex G -invariant G -minimal subset X ′ 6 = ∅ whih has no Is( X ′ ) -xe d p oint at innity. Consider the immediate orollary . Corollary 3.12. L et X b e a pr op er CA T(0) sp a e suh that G = Is( X ) ats o omp atly and minimal ly. If G ontains a nitely gener ate d latti e, then G has no xe d p oint at innity. This sho ws that the mere existene of a nitely generated lattie imp oses restritions on o ompat CA T(0) spaes; m u h more detailed results in that spirit will b e giv en in Setion 6 . W e do not kno w whether the statemen t of Corollary 3.12 remains true without the nite generation assumption on the lattie (see Problem 7.3 b elo w). Example 3.13 . W e emphasise that the full isometry group of a o ompat prop er CA T(0) spae ma y ha v e global xed p oin ts at innit y; in fat, the spae migh t ev en b e homogeneous, as it is the ase for E. Hein tze's manifolds [ Hei74 ℄ men tioned earlier. An ev en simpler w a y to onstrut o ompat prop er CA T(0) spae with this prop ert y is to mimi Example 7.6 in [ CM08 ℄: Start from a regular tree T , assuming for deniteness that the v aleny is three. Replae ev ery v ertex b y a ongruen t op y of an isoseles triangle that is not equilateral, in su h a w a y that its distinguished v ertex alw a ys p oin ts to a xed p oin t at innit y (of the initial tree). Then the stabiliser H in Is( T ) of that p oin t at innit y still ats faithfully and o ompatly on the mo died spae T ′ ; the onstrution is so that the isometry group of T ′ is in fat redued to H . 16 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD W e shall also establish a strengthening of Corollary 3.12 , whi h an b e view ed as a form of Borel (or geometri) densit y theorem without assumption ab out xed p oin ts at innit y . Theorem 3.14. L et X b e a pr op er CA T(0) sp a e suh that G = Is( X ) ats o omp atly and minimal ly. Assume ther e is a nitely gener ate d latti e Γ < G . Then Γ ats minimal ly on X and mor e over al l Γ -xe d p oints at innity ar e ontaine d in the b oundary of the (p ossibly trivial) Eulide an fator of X . W e no w turn to the pro ofs. In the ase of a disrete o ompat group Γ = G , a v ersion of the follo wing w as rst established b y BurgerS hro eder [ BS87 ℄ (as p oin ted out in [ AB98 , Corollary 2.7℄). Prop osition 3.15. L et X b e a pr op er CA T(0) sp a e, G < I s ( X ) a lose d sub gr oup whose ation on X is o omp at and Γ < G a nitely gener ate d latti e. Then ther e exists a Γ - invariant lose d onvex subset Y ⊆ X whih splits Γ -e quivariantly as Y = E × W , wher e E is a (p ossibly 0 -dimensional) Eulide an sp a e on whih Γ ats by tr anslations and suh that ∂ E ontains the xe d p oint set of G in ∂ X . Pr o of. W e an assume that there are G -xed p oin ts at innit y , sine otherwise there is nothing to pro v e. W e laim that for an y G -xed p oin t ξ there is a geo desi line σ : R → X with σ (+ ∞ ) = ξ su h that an y γ ∈ Γ mo v es σ to within a b ounded distane of itself and hene to a parallel line b y on v exit y of the metri. Indeed, let c : R + → X b e a geo desi ra y with c ( ∞ ) = ξ and let { γ i } b e as in Lemma 2.11 . Then, b y ArzelàAsoli, there is a subsequene I ⊆ N and a geo desi line σ : R → X su h that σ ( t ) = lim i ∈ I γ i c ( t + i ) for all t . Sine ea h g ∈ G has b ounded displaemen t along c , the sequene { γ i g γ − 1 i } i ∈ I is b ounded and th us w e an assume that it on v erges for all g (realling that G is seond oun table, but w e shall only onsider g ∈ Γ an yw a y). Sine Γ is disrete and nitely generated, w e an further restrit I so that there is γ ∞ ∈ Γ su h that γ i γ γ − 1 i = γ ∞ γ γ − 1 ∞ ∀ γ ∈ Γ , i ∈ I . Sine d ( γ γ − 1 ∞ σ ( t ) , γ − 1 ∞ σ ( t )) = lim i ∈ I d ( γ − 1 i γ ∞ γ γ − 1 ∞ γ i c ( t + i ) , c ( t + i )) = lim i ∈ I d ( γ c ( t + i ) , c ( t + i )) ≤ d ( γ c (0) , c (0)) , It no w follo ws that ev ery γ ∈ Γ has b ounded displaemen t length along the geo desi γ − 1 ∞ σ . Th us the same holds for the geo desi σ whi h is therefore (b y on v exit y) translated to a parallel line b y ea h elemen t of Γ as laimed. Consider a at E ⊆ X that is maximal for the prop ert y that ea h elemen t of Γ has onstan t displaemen t length on E . Let Y b e the union of all ats that are at nite distane from E . One sho ws that Y splits as Y ∼ = E × W for some losed on v ex W ⊆ X using the Sandwi h Lemma [ BH99 , I I.2.12℄ and Lemma I I.2.15 of [ BH99 ℄ just lik e in Setion 3.B in [ CM08 ℄. The denition of Y sho ws that Γ preserv es Y as w ell as its splitting and ats on the E o ordinate b y translations. It remains to sho w that an y G -xed p oin t ξ ∈ ∂ X b elongs to ∂ E . First, ξ ∈ ∂ Y sine ∂ Y = ∂ X b y Corollary 2.10 ; w e th us represen t ξ b y a ra y c : R + → Y . Let no w σ b e a geo desi line as pro vided b y the laim. W e an assume that σ lies in Y b eause it w as onstruted from Γ -translates of c and Y is Γ -in v arian t. One an write σ = ( σ E , σ W ) where σ E , σ W are linearly re-parametrised geo desis in E and W , see [ BH99 , I.5.3℄. W e need to pro v e that σ W has zero sp eed. Sine an y giv en γ ∈ Γ has onstan t displaemen t along σ and on ea h of the parallel opies of E individually , its displaemen t is onstan t on the union of ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 17 all parallel opies of E visited b y σ , whi h is E × σ W ( R ) . The latter b eing again a at, the maximalit y of E sho ws that σ W is onstan t. Pr o of of The or em 3.11 . Let Y = E × W ⊆ X b e as in Prop osition 3.15 . Reall that ∂ Y = ∂ X b y Corollary 2.10 . W e laim that ∂ X has irumradius > π / 2 . Indeed, it w ould otherwise ha v e a G -xed irumen tre b y Prop osition 3.1 in [ CM08 ℄, but this irumen tre annot b elong to ∂ E sine E is Eulidean; this on tradits Prop osition 3.15 . W e no w apply Corollary 3.10 in [ CM08 ℄. This yields a anonial G -in v arian t losed on v ex subset X ′ , whi h is minimal with resp et to the prop ert y that ∂ X ′ = ∂ X . It follo ws in partiular b y Corollary 2.10 that Γ ats minimally on X ′ . Let no w X ′ = E ′ × X ′ 0 b e the anonial splitting, where E ′ is the maximal Eulidean fator [ BH99 , I I.6.15℄. On the one hand, sine X ′ is Γ -minimal, Prop osition 3.15 applied to X ′ sho ws that G has no xed p oin ts in ∂ X ′ 0 sine E ′ is maximal as a Eulidean fator. On the other hand, Is( E ′ ) xes no p oin t at innit y on E ′ . W e dedue that Is( X ′ ) ∼ = Is( E ′ ) × Is( X ′ 0 ) has indeed no xed p oin t at innit y . End of pr o of of The or em 3.14 . Arguing as in the pro of of Theorem 3.11 , w e establish that X is Γ -minimal. Let X = X ′ × E b e the anonial splitting, where E is the maximal Eulidean fator. Sine an y isometry of X deomp oses uniquely as isometries of E and X ′ (I I.6.15 in [ BH99 ℄), is sues to sho w that Γ has no xed p oin t in ∂ X ′ . This follo ws from Prop osition 2.1 applied to the G -ation on X ′ . End of pr o of of The or em 3.8 . Assume rst that X is G -minimal, realling that this is the ase if X is geo desially omplete b y Lemma 3.13 in [ CM08 ℄. In view of Corollary 3.12 , w e an apply Theorem 1.6 in [ CM08 ℄ and w e are therefore in the setting of Prop osition 3.6 . Sine the group Γ A pro vided b y that prop osition on tains a nite index subgroup isomorphi to Z n , w e ha v e already established ( i ) under the additional minimalit y assumption. In order to sho w ( ii ), it sues b y Remark 3.7 to pro v e that an y ompat subgroup of G normalised b y Γ is trivial. This follo ws from the fat that X is Γ -minimal, as established in Theorem 3.14 . It remains to pro v e ( i ) without the assumption that X is G -minimal. Let Y ⊆ X b e the G -minimal set pro vided b y Theorem 3.11 and let Y ∼ = R m × Y ′ b e its Eulidean deomp osition. Then m ≥ n b eause of the haraterisation of the Eulidean fator in terms of Cliord isometries [ BH99 , I I.6.15℄; indeed, an y (non-trivial) Cliord isometry of X restrits non-trivially to Y b eause Y has nite o-diameter. The k ernel F ✁ Γ of the Γ -ation on Y is nite and th us w e an assume that it is en tral up on replaing Γ with a nite index subgroup. P assing to a further nite index subgroup, w e kno w from the minimal ase that Γ /F splits as Γ /F = Z m × Λ ′ . Let Γ Z m , Γ ′ ✁ Γ b e the pre-images in Γ of those fators. Th us w e an write Γ = Γ Z m · Γ ′ with Γ Z m ∩ Γ ′ ⊆ F . It is straigh tforw ard that a nite en tral extension of Z m is virtually Z m (see e.g. [ BH99 , I I.7.9℄). Therefore Γ on tains a nite index subgroup isomorphi to Z m × Γ ′ and the result follo ws sine m ≥ n . Pr o of of Cor ol lary 3.10 . Notie that a splitting Γ 0 ∼ = Z n × Γ ′ with Γ 0 normal and n maximal pro vides a normal subgroup Z n ✁ Γ sine Z n is harateristi in Γ 0 . Therefore, giv en Theorem 3.8 , it only remains to see that a normal Z n ✁ Γ of maximal rank fores X to ha v e a Eulidean fator of dimension at least n . Otherwise, the pro jetion of Γ to the non- Eulidean fator X ′ w ould b e a lattie b y Theorem 3.8 ( ii ) and on tain an innite normal amenable subgroup, on traditing Corollary 2.7 ( i ). 18 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Finally , w e reord that Theorem 1.3 is on tained in Theorem 3.8 and Corollary 3.10 for (i), and Corollary 3.12 and Theorem 3.14 for (ii). 4. CA T(0) la tties, I I: pr oduts 4.A. Irreduible latties in CA T(0) spaes. Reall from that a (top ologial) group is alled irreduible if no (op en) nite index subgroup splits non-trivially as a diret pro dut of (losed) subgroups. F or example, an y lo ally ompat group ating on tin uously , prop erly , minimally , without xed p oin t at innit y on an irreduible prop er CA T(0) spae is irreduible b y Theorem 1.10 in [ CM08 ℄. In partiular, an abstrat group Γ is irreduible if it do es not virtually split. This ter- minology is inspired b y the onept of irreduibilit y for losed manifolds, whi h means that no nite o v er of the manifold splits non-trivially . Of ourse, the univ ersal o v er of su h a manifold an still split. Indeed, one gets man y lassial CA T(0) groups b y onsidering irre- duible latties in pro duts of simple Lie groups or more generally of semi-simple algebrai groups o v er v arious lo al elds. The latter onept of irreduibilit y for latties is dened as follo ws: A lattie Γ < G = G 1 × · · · × G n in a pro dut of lo ally ompat groups is alled an irreduible lattie if its pro jetions to an y subpro dut of the G i 's are dense and ea h G i is non-disrete. The p oin t of this notion (and of the nearly onfusing terminology) is that it prev en ts Γ and its nite index subgroups from splitting as a pro dut of latties in G i . Moreo v er, if all G i 's are en tre-free simple Lie (or algebrai) groups without ompat fators, the irreduibilit y of Γ as a lattie is equiv alen t to its irreduibilit y as a group in and for itself; this is a result of Margulis [ Mar91 , I I.6.7℄. As w e shall see in Theorem 4.2 b elo w, a v ersion of this equiv alene holds for latties in the isometry group of a CA T(0) spae. Remark 4.1. (i) The non-disreteness of G i is often omitted from this denition; the dierene is inessen tial sine the notion of a lattie is trivial for disrete groups. Notie ho w ev er that our denition ensures that all G i are non-ompat and that n ≥ 2 . (ii) One v eries that any lattie Γ < G = G 1 × G 2 is an irreduible lattie in the pro dut G ∗ < G of the losures G ∗ i < G i of its pro jetions to G i (pro vided these pro jetions are non-disrete). The follo wing geometri v ersion of Margulis' riterion on tains Theorem 1.4 from the In tro dution. Theorem 4.2. L et X b e a pr op er CA T(0) sp a e, G < Is( X ) a lose d sub gr oup ating o omp atly on X , and Γ < G a nitely gener ate d latti e. (i) If Γ is irr e duible as an abstr at gr oup, then for any nite index sub gr oup Γ 0 < Γ and any Γ 0 -e quivariant splitting X = X 1 × X 2 with X 1 and X 2 non- omp at, the pr oje tion of Γ 0 to b oth Is( X i ) is non-disr ete. (ii) If in addition the G -ation is minimal, then the onverse statement holds as wel l. Remark 4.3. Reall that the G -minimalit y is automati if X is geo desially omplete (Lemma 3.13 in [ CM08 ℄). Statemen t (ii) fails ompletely without minimalit y (as witnessed for instane b y the unosmop olitan mien of an equiv arian t mane). Pr o of of The or em 4.2 . Supp ose Γ irreduible. Let X ′ ⊆ X b e the anonial subspae pro- vided b y Theorem 3.11 . By Theorem 3.8 , the spae X ′ has no Eulidean fator unless X = R and Γ is virtually yli, in whi h ase the desired statemen t is empt y . ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 19 W e rst deal with the ase when G ats minimally on X ; b y Theorem 2.4 this amoun ts to assume X = X ′ . Supp ose for a on tradition that for Γ 0 and X ′ = X ′ 1 × X ′ 2 as in the statemen t, the pro jetion G 1 of Γ 0 to Is( X ′ 1 ) is disrete. Let G 2 b e the losure of the pro jetion of Γ 0 to Is( X ′ 2 ) and notie that b oth G i are ompatly generated sine Γ and hene also Γ 0 is nitely generated. The pro jetion Γ 2 of Γ 0 ∩ (1 × G 2 ) to G 2 is a lattie (b y Lemma 3.2 or b y Prop osition 3.1 ); b eing normal, it is o ompat and hene nitely generated. By Theorem 3.14 , the group Γ 0 , and hene also G 2 , ats minimally and without xed p oin t at innit y on X ′ 2 . Therefore Corollary 2.7 ( ii ) implies that the en traliser Z G 2 (Γ 2 ) is trivial. But Γ 2 is disrete, normal in G 2 , and nitely generated. Hene Z G 2 (Γ 2 ) is op en and th us G 2 is disrete. Therefore, the pro dut G 1 × G 2 , whi h on tains Γ 0 , is a lattie in Is( X ′ 1 ) × Is( X ′ 2 ) and th us in G . No w the index of Γ 0 in G 1 × G 2 is nite and th us Γ 0 splits virtually , a on tradition. W e no w ome ba k to the general ase X ′ ⊆ X and supp ose that X p ossesses a Γ 0 - equiv arian t splitting X = X 1 × X 2 . The group H = Is( X 1 ) × I s ( X 2 ) < Is( X ) on tains Γ 0 ; hene its ation on X ′ is minimal without xed p oin t at innit y b y Corollary 3.12 . There- fore, the splitting theorem [ Mon06 , Theorem 9℄ implies that X ′ p ossesses a Γ 0 -equiv arian t splitting X ′ = X ′ 1 × X ′ 2 indued b y X = X 1 × X 2 via H . Up on replaing Γ 0 b e a nite index subgroup, the preeding paragraph th us yields a splitting Γ 0 /F ∼ = G 1 × G 2 of the image of Γ 0 in Is( X ′ ) , where F denotes the k ernel of the Γ 0 -ation on X ′ . Sine F is nite, so is the pro jetion to Is( X 3 − i ) of the preimage b G i of G i in Γ , for i = 1 , 2 . Therefore up on passing to a nite index subgroup w e ma y and shall assume that b G i ats trivially on Is( X 3 − i ) . No w the subgroup of Is( X 1 ) × Is ( X 2 ) generated b y b G 1 and b G 2 splits as b G 1 × b G 2 and is ommensurable to Γ 0 , a on tradition. Con v ersely , supp ose no w that the G -ation is minimal and that Γ = Γ ′ × Γ ′′ splits non- trivially (after p ossibly ha ving replaed it b y a nite index subgroup). If X = R n , then reduibilit y of Γ fores n ≥ 2 and w e are done in view of the struture of Bieb erba h groups. If X is not Eulidean but has a Eulidean fator, then Theorem 3.8 ( ii ) pro vides a disrete pro jetion of Γ to the non-Eulidean fator Is( X ′ ) ; furthermore, X ′ is indeed non-ompat as desired sine otherwise b y minimalit y it is redued to a p oin t, on trary to our assumption. If on the other hand X has no Eulidean fator, then Γ ats minimally and without xed p oin t at innit y b y Theorem 3.11 . Then the desired splitting is pro vided b y the splitting theorem [ Mon06 , Theorem 9℄. Both pro jetions of Γ are disrete, indeed isomorphi to Γ ′ resp etiv ely Γ ′′ b eause the ited splitting theorem ensures omp onen t wise ation of Γ . W e no w briey turn to uniquely geo desi spaes and to the analogues in this setting of some of P . Eb erlein's results for Hadamard manifolds. Theorem 4.4. L et X b e a pr op er CA T(0) sp a e with uniquely extensible ge o desis suh that Is( X ) ats o omp atly on X . If Is( X ) admits a nitely gener ate d non-uniform irr e duible latti e, then X is a symmetri sp a e (without Eulide an fator). Pr o of. The ation of Is( X ) is minimal b y Lemma 3.13 in [ CM08 ℄ and without xed p oin t at innit y b y Corollary 3.12 . Th us w e an apply Theorem 1.6 in [ CM08 ℄ and A ddendum 1.8 in [ CM08 ℄. Notie that Is( X ) itself is non-disrete sine it on tains a non-uniform lattie; moreo v er, if it admits more than one fator in the deomp osition of Theorem 1.6 in [ CM08 ℄, then the latter are all non-disrete b y Theorem 4.2 . Therefore, w e an apply Theorem 7.10 20 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD in [ CM08 ℄ to all fators of X . It remains only to justify that X has no Eulidean fa- tor; otherwise, Auslander's theorem (ompare also Theorem 3.8 ) implies X = R , whi h is inompatible with the fat that Γ is non-uniform. The follo wing related result is due to P . Eb erlein in the manifold ase (Prop osition 4.5 in [ Eb e82 ℄). W e shall establish another result of the same v ein later without assuming that geo desis are uniquely extensible (see Theorem 6.6 ). Theorem 4.5. L et X b e a pr op er CA T(0) sp a e with uniquely extensible ge o desis and Γ < Is( X ) b e a disr ete o omp at gr oup of isometries. If Γ is irr e duible (as an abstr at gr oup) and X is r e duible, then X is a symmetri sp a e (without Eulide an fator). Pr o of. One follo ws line-b y-line the pro of of Theorem 4.4 . The only dierene is that, in the presen t on text, the non-disreteness of the isometry group of ea h irreduible fator of X follo ws from Theorem 4.2 sine X is assumed reduible. W e an no w onlude the pro of of Theorem 1.10 from the In tro dution. The rst state- men t w as established in Theorem 7.10 in [ CM08 ℄. The seond follo ws from Theorem 4.4 and the third from Theorem 4.5 in the uniform ase, and from Theorem 4.4 in the non-uniform one. 4.B. The h ull of a lattie. Let X b e a prop er CA T(0) spae X su h that Is( X ) ats o ompatly on X . Let Γ < Is( X ) b e a nitely generated lattie; note that the ondition of nite generation is redundan t if Γ is o ompat b y Lemma 3.3 . Theorem 3.11 pro vides a anonial Is( X ) -in v arian t subspae X ′ ⊆ X su h that G = I s ( X ′ ) has no xed p oin t at innit y . In this setion w e shall dene the h ull H Γ of the lattie Γ ; this is a lo ally ompat group H Γ < Is( X ′ ) anonially atta hed to the situation and on taining the image of Γ in Is( X ′ ) . F or simpliit y , w e rst treat the sp eial ase where Is( X ) ats minimally; th us X ′ = X and G = I s( X ) . Applying Theorem 1.6 in [ CM08 ℄ and A ddendum 1.8 in [ CM08 ℄, w e see in partiular that Γ p ossesses a anonial nite index normal subgroup Γ ∗ = Γ ∩ G ∗ whi h is the k ernel of the Γ -ation b y p erm utation on the set of fators in the deomp osition giv en b y A ddendum 1.8 in [ CM08 ℄. In the lassial ase when X is a symmetri spae, the losure of the pro jetion of Γ to the isometry group Is( X i ) of ea h fator is an op en subgroup of nite index , as so on as X is reduible. This is no longer true in general, ev en in the ase of Eulidean buildings. In fat, the same Γ ma y (and generally do es) o ur as lattie in inreasingly large am bien t groups Γ < G < G ′ < G ′′ < · · · . In order to address this issue, w e dene the h ull as follo ws. Consider the losed subgroup H Γ ∗ < G whi h is the diret pro dut of the losure of the images of Γ ∗ in ea h of the fators in the deomp osition of Theorem 1.6 in [ CM08 ℄. Then set H Γ = Γ · H Γ ∗ . In other w ords, w e ha v e inlusions Γ < H Γ < G. The losed subgroup H Γ ∗ is nothing but the h ull of the lattie Γ ∗ . It oinides with H ∗ Γ = H Γ ∩ G ∗ . In partiular H ∗ Γ = H Γ ∗ is a diret pro dut of irreduible groups satisfying all the restritions of Theorem 1.10 in [ CM08 ℄ (exept for the p ossible Eulidean motion fator), and the image of Γ ∗ in ea h of these fators is dense. Remark 4.6. Notie that Γ is alw a ys a lattie in H Γ (b y [ Rag72 , Lemma 1.6℄). W e emphasise that H Γ is non-disrete and that Γ ∗ is an irreduible lattie in H Γ ∗ (in the ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 21 sense of 4.A ) as so on as Γ is irreduible as a group and X is reduible; this follo ws from Theorem 4.2 . W e no w dene the h ull H Γ < G in the general situation G = Is( X ′ ) with X ′ ⊆ X giv en b y Theorem 3.11 . Sine Is( X ) \ X is o ompat, it follo ws that X ′ is r -dense in X for some r > 0 and the anonial map Is( X ) → G is prop er. Let F Γ ✁ Γ b e the nite k ernel of the indued map Γ → G and write Γ ′ := Γ /F Γ . Then the h ull of Γ is dened b y H Γ := H Γ ′ (reduing to the ab o v e ase). In other w ords, Γ sits in H Γ only mo dulo the anonial nite k ernel F Γ . In fat, F Γ is ev en anonially atta hed to Γ view ed as an abstrat group. Lemma 4.7. F Γ is a (ne essarily unique) maximal nite normal sub gr oup of Γ . Mor e over, X ′ is Γ ′ -minimal. Pr o of. The Γ ′ -ation on X ′ is minimal b y an appliation of Theorem 3.14 and therefore ev ery nite normal subgroup of Γ ′ is trivial. Sine moreo v er the Γ -ation on X ′ is prop er, it follo ws that a normal subgroup of Γ is nite if and only if it lies in F Γ . F or later referenes, w e reord the follo wing exp eted fat. Lemma 4.8. Assume that Γ is irr e duible. If X ′ is r e duible, then H Γ ontains the identity omp onent of G := Is( X ′ ) . In fat ( H Γ ) ◦ = G ◦ is a semi-simple Lie gr oup with trivial entr e and no omp at fator. Pr o of. By Theorem 3.8 , the h yp otheses on Γ imply that X ′ has no Eulidean fator. Th us ea h almost onneted fator of G ∗ is a simple Lie group with trivial en tre and no ompat fator. The pro jetion of Γ ∗ to ea h of these fators is non-disrete b y Theorem 4.2 and the assumption made on X ′ . Its losure is semi-simple and Zariski dense b y Theorem 2.4 and Prop osition 2.8 . The result follo ws. 4.C. On the anonial disrete k ernel. Let G = G 1 × G 2 b e a lo ally ompat group and Γ < G b e an irreduible lattie. It follo ws from irreduibilit y that the pro jetion to G i of the k ernel of the pro jetion Γ → G j 6 = i is a normal subgroup of G i . In other w ords, w e ha v e a anonial disrete normal subgroup Γ i ✁ G i dened b y Γ 1 = Pro j G 1 Γ ∩ ( G 1 × 1) (and lik ewise for Γ 2 ) whi h w e all the anonial disrete k ernel of G i (dep ending on Γ ). W e observ e that the image Γ = Γ / (Γ 1 · Γ 2 ) of Γ in the anonial quotien t G 1 / Γ 1 × G 2 / Γ 2 is still an irreduible lattie (see Prop osi- tion 3.1 (ii)) and has the additional prop ert y that it pro jets injetiv ely in to b oth fators. In this subsetion, w e ollet some basi fats on latties in (pro duts of ) totally dison- neted lo ally ompat groups, adapting ideas of M. Burger and Sh. Mozes (see Prop osi- tions 2.1 and 2.2 in [ BM00b ℄). Prop osition 4.9. L et Γ < G = G 1 × G 2 b e an irr e duible latti e. Assume that G 2 is total ly dis onne te d, omp atly gener ate d and without non-trivial omp at normal sub gr oup. If Γ is r esidual ly nite, then anoni al the disr ete kernel Γ 2 = Γ ∩ (1 × G 2 ) ommutes with the disr ete r esidual G ( ∞ ) 2 . Reall that the disrete residual G ( ∞ ) of a top ologial group G is b y denition the in tersetion of all op en normal subgroups. It is imp ortan t to remark that, b y Corollary 6.13 in [ CM08 ℄ the disrete residual of a non-disrete ompatly generated lo ally ompat group without non-trivial ompat normal subgroup is neessarily non-trivial. 22 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Pr o of of Pr op osition 4.9 . By a sligh t abuse of notation, w e shall iden tify G 2 with the sub- group 1 × G 2 of G . Giv en a nite index normal subgroup Γ 0 ✁ Γ , the in tersetion Γ 0 ∩ Γ 2 is a disrete normal subgroup of G 2 (b y irreduibilit y), on tained as a nite index subgroup in Γ 2 . Th us G 2 ats b y onjugation on the nite quotien t Γ 2 / Γ 0 ∩ Γ 2 . In partiular the k ernel of this ation is a nite index losed normal subgroup, whi h is th us op en. Therefore, the disrete residual G ( ∞ ) 2 ats trivially on Γ 2 / Γ 0 ∩ Γ 2 . In other w ords, this means that [Γ 2 , G ( ∞ ) 2 ] ⊆ Γ 0 ∩ Γ 2 . Assume no w that Γ is residually nite. The preeding argumen t then sho ws that the omm utator [Γ 2 , G ( ∞ ) 2 ] is trivial, as desired. Prop osition 4.10. L et Γ < G = G 1 × G 2 b e a o omp at latti e in a pr o dut of omp atly gener ate d lo al ly omp at gr oups. Assume that G 2 is total ly dis onne te d and that the en- tr aliser in G 1 of any o omp at latti e of G 1 is trivial. If the disr ete kernel Γ 2 = Γ ∩ (1 × G 2 ) is trivial, then the quasi- entr e Q Z ( G 2 ) is top olo gi al ly lo al ly nite. Pr o of. Let S ⊆ Q Z ( G 2 ) b e a nite subset of the quasi-en tre. Then G 2 p ossesses a ompat op en subgroup U whi h en tralises S . By Lemma 3.2 the group Γ U = Γ ∩ ( G 1 × U ) is a o ompat lattie in G 1 × U . In partiular, there is a nite generating set T ⊆ Γ U . By a lemma of Selb erg [ Sel60 ℄, the group Z Γ ( T ) is a o ompat lattie in Z G ( T ) . But Z G ( T ) = Z G (Γ U ) ⊆ 1 × G 2 sine the pro jetion of Γ U to G 1 is a o ompat lattie. Sine the disrete k ernel Γ ∩ (1 × G 2 ) is trivial b y h yp othesis, the en traliser Z Γ ( T ) is trivial and, hene, Z G ( T ) is ompat. By onstrution S is on tained in Z G ( T ) , whi h yields the desired result. 4.D. Residually nite latties. Theorem 4.11. L et X b e a pr op er CA T(0) sp a e suh that Is( X ) ats o omp atly and minimal ly. L et Γ < Is ( X ) b e a nitely gener ate d latti e. Assume that Γ is irr e duible and r esidual ly nite. Then we have the fol lowing (se e Se tion 4.B for the notation): (i) Γ ∗ ats faithful ly on e ah irr e duible fator of X . (ii) If Γ is o omp at and X is r e duible, then for any lose d sub gr oup G < Is( X ) ontaining H Γ ∗ , we have Q Z ( G ) = Q Z ( G ∗ ) = 1 . F urthermor e so c( G ∗ ) is a dir e t pr o dut of r non-disr ete lose d sub gr oups, e ah of whih is har ateristi al ly simple, wher e r is the numb er of irr e duible fators of X . (W e emphasise that the irreduibilit y assumption onerns Γ as an abstrat group; ompare ho w ev er Remark 4.6 .) Pr o of. If X is irreduible, there is nothing to pro v e. W e assume heneforth that X is reduible. In view of Theorem 3.8 , X has no Eulidean fator. Moreo v er, Corollary 3.12 implies that Is( X ) xes no p oin t at innit y . In partiular, Γ and H Γ ∗ at minimally without xed p oin t at innit y b y Theorem 2.4 . Let H 1 , . . . , H r b e the irreduible fators of H Γ ∗ ; th us r oinides with the n um b er of irreduible fators of X . In view of Theorem 4.2 , the group Γ ∗ is an irreduible lattie in this pro dut. By Corollary 1.11 in [ CM08 ℄ and Theorem 2.4 , ea h H i is either a en tre-free simple Lie group or totally disonneted with trivial amenable radial. If H 1 is a simple Lie group, then it has no non-trivial disrete normal subgroup and hene (Γ ∗ ) 1 := Γ ∗ ∩ ( H 1 × 1 × · · · × 1) = 1 . ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 23 If H 1 is totally disonneted, then b y Prop osition 4.9 the anonial disrete k ernel (Γ ∗ ) 1 omm utes with the disrete residual H ( ∞ ) 1 , whi h is non-trivial b y Corollary 6.13 in [ CM08 ℄. Th us Z H 1 ( H ( ∞ ) 1 ) = 1 b y Theorem 1.10 in [ CM08 ℄ and hene (Γ ∗ ) 1 = 1 . Assertion (i) no w follo ws from a straigh tforw ard indution on r . Assume next that Γ is o ompat. Let G 1 , . . . , G r b e the irreduible fators of G ∗ . By Lemma 4.8 and Prop osition 4.10 , and in view of P art (i), for ea h totally disonneted fa- tor G i , the quasi-en tre Q Z ( G i ) is top ologially lo ally nite. Its losure is th us amenable, hene trivial b y Theorem 1.10 in [ CM08 ℄. Moreo v er, the quasi-en tre of ea h almost on- neted fator is trivial as w ell b y Lemma 4.8 . Clearly the pro jetion of the quasi-en tre of G ∗ to the irreduible fator G i is on tained in Q Z ( G i ) . This sho ws that Q Z ( G ∗ ) is trivial. Hene so is Q Z ( G ) , sine it on tains Q Z ( G ∗ ) as a nite index subgroup and sine G has no non-trivial nite normal subgroup b y Corollary 5.8 in [ CM08 ℄. No w the desired onlusion follo ws from Prop osition 6.11 in [ CM08 ℄. 4.E. Strong rigidit y for pro dut spaes. In [ CM08 ℄, w e presen ted a few sup errigidit y results (Setion 8 in [ CM08 ℄). Sup errigidit y should on tain, in partiular, strong rigidit y à la Mosto w. This is indeed the on ten t of Theorem 4.12 b elo w, where an isomorphism of latties is sho wn to extend to an am bien t group. Ho w ev er, in on trast to the lassial ase of symmetri spaes, whi h are homogeneous, the full isometry group do es not in general determine the spae sine CA T(0) spaes are in general not homogeneous. Another dierene is that the h ull of a lattie, as desrib ed in Setion 4.B , is generally smaller than the full isometry group of the am bien t CA T(0) spae. In view of the denition of the h ull, the follo wing statemen t is non-trivial only when X (or an in v arian t subspae) is reduible; this is exp eted sine w e w an t to use sup errigidit y for irreduible latties in pro duts. Theorem 4.12. L et X, Y b e pr op er CA T(0) sp a es and Γ , Λ disr ete o omp at gr oups of isometries of X , r esp e tively Y , not splitting (virtual ly) a Z n fator. Then any isomorphism Γ ∼ = Λ determines an isomorphism H Γ ∼ = H Λ suh that the fol- lowing ommutes: Γ / / Λ H Γ ∼ = / / H Λ Theorem 4.12 pro vides a partial answ er to Question 21 in [ FHT08 ℄. Remark 4.13. The assumption on Z n fators is equiv alen t to exluding Eulidean fa- tors from X (or its anonial in v arian t subspae) b y Theorem 3.8 . On the one hand, this assumption is really neessary for the theorem to hold, ev en for symmetri spaes, sine one an t wist the pro dut using a Γ -ation on the Eulidean fator when H 1 (Γ) 6 = 0 (om- pare [ L Y72 , 4℄). On the other hand, sine Bieb erba h groups are ob viously Mosto w-rigid, Theorem 4.12 together with Theorem 3.8 giv e us as omplete as p ossible a desription of the situation with Z n fators. Pr o of. Let X ′ ⊆ X b e the subset pro vided b y Theorem 3.11 . W e retain the notation F Γ ✁ Γ and Γ ′ = Γ /F Γ < Is( X ′ ) from Setion 4.B and reall from Lemma 4.7 that F Γ dep ends only on Γ as abstrat group and that X ′ is Γ ′ -minimal. W e dene Y ′ , F Λ and Λ ′ in the same w a y and ha v e the orresp onding lemma. In partiular, it follo w that the isomorphism Γ ∼ = Λ 24 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD desends to Γ ′ ∼ = Λ ′ . Therefore, w e an and shall assume from no w on that X and Y are minimal and Γ < H Γ < Is( X ) , Λ < H Λ < Is( Y ) . By Theorem 3.8 , w e kno w that X, Y ha v e no Eulidean fator. Th us Γ , Λ ha v e no xed p oin t at innit y b y Theorem 3.14 . W e laim that Γ has a nite index subgroup Γ † whi h deomp oses as a pro dut Γ † = Γ 1 × · · · × Γ s of irreduible fators, with s maximal for this prop ert y . Indeed, otherwise w e ould apply the splitting theorem of [ Mon06 ℄ to a hain a nite index subgroups and on tradit the prop erness of X . W e write Λ † = Λ 1 × · · · × Λ s for the orresp onding groups in Λ . Com bining the splitting theorem with A ddendum 1.8 in [ CM08 ℄, it follo ws from the denition of the h ull that it is suien t to pro v e the statemen t for s = 1 . W e assume heneforth that Γ , and hene also Λ , is irreduible. F urthermore, if X and Y are b oth irreduible, then H Γ = Γ and H Λ = Λ and the desired statemen t is empt y . W e no w assume that X is reduible. By Theorem 2.4 , the lattie Γ (resp. Λ ) ats minimally without xed p oin t at innit y on X (resp. Y ). Theorem 8.4 in [ CM08 ℄ yields a on tin uous morphism f : H Γ ∗ → H Λ ∗ , whi h sho ws in partiular (b y the splitting theorem [ Mon06 ℄) that Y is reduible as w ell. A seond appliation of Theorem 8.4 in [ CM08 ℄ yields a seond on tin uous morphism f ′ : H Λ ∗ → H Γ ∗ . Notie that the resp etiv e restritions to Γ ∗ and Λ ∗ oinides with the giv en isomorphism and its in v erse. In partiular f ′ ◦ f (resp. f ′ ◦ f ) is the iden tit y on Γ (resp. Λ ). By denition of the h ull, it follo ws that f ′ ◦ f (resp. f ′ ◦ f ) is the iden tit y on H Γ ∗ (resp. H Λ ∗ ). The desired result nally follo ws, sine there is a anonial isomorphism Γ / Γ ∗ ∼ = H Γ /H Γ ∗ and sine the ation of H Γ /H Γ ∗ on H Γ ∗ is anonially determined b y the ation of Γ / Γ ∗ on Γ ∗ . The ab o v e pro of sho ws in partiular that amongst spaes that are Γ -minimal without Eulidean fator, the n um b er of irreduible fators dep ends only up on the group Γ . If w e om bine this with Theorem 3.14 , Corollary 3.10 and Theorem 3.8 ( ii ), w e obtain that the n um b er of fators in the de Rham deomp osition (4.i) X ′ ∼ = X 1 × · · · × X p × R n × Y 1 × · · · × Y q of A ddendum 1.8 in [ CM08 ℄ is an in v arian t of the group: Corollary 4.14. L et X b e a pr op er CA T(0) sp a e and Γ < Is( X ) b e a gr oup ating pr op erly dis ontinuously and o omp atly. Then any other suh sp a e admitting a pr op er o omp at Γ -ation has the same numb er of fators in ( 4.i ) and the Eulide an fator has same dimension. (W e reall that minimalit y is automati when X is geo desially omplete: Lemma 3.13 in [ CM08 ℄.) 5. Arithmetiity of abstra t la tties The main goal of this setion is to pro v e Theorem 1.9 , whi h w e no w state in a sligh tly more general form. F ollo wing G. Margulis [ Mar91 , IX.1.8℄, w e shall sa y that a simple algebrai group G dened o v er a eld k is admissible if none of the follo wing holds: c har( k ) = 2 and G is of t yp e A 1 , B n , C n or F 4 . c har( k ) = 3 and G is of t yp e G 2 . A semi-simple group will b e said admissible if all its fators are. Theorem 5.1. L et Γ < G = G 1 × · · · × G n b e an irr e duible nitely gener ate d latti e, wher e e ah G i is any lo al ly omp at gr oup. If Γ admits a faithful Zariski-dense r epr esentation in an admissible semi-simple gr oup (over any eld), then the amenable r adi al R of G is omp at and the quasi- entr e Q Z ( G ) ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 25 is virtual ly ontaine d in Γ · R . F urthermor e, up on r eplaing G by a nite index sub gr oup, the quotient G/R splits as G + × Q Z ( G/R ) wher e G + is a semi-simple algebr ai gr oup and the image of Γ in G + is an arithmeti latti e. Sine the pro jetion map G → G/R is prop er, the statemen t of Theorem 5.1 implies in partiular that Q Z ( G/R ) is disrete. Corollary 5.2. L et G = G 1 × · · · × G n b e a pr o dut of lo al ly omp at gr oups. Assume that G admits a nitely gener ate d irr e duible latti e with a faithful Zariski-dense r epr esentation in a semi-simple gr oup over some eld of har ateristi 6 = 2 , 3 . Then G is a omp at extension of a dir e t pr o dut of a semi-simple algebr ai gr oup by a disr ete gr oup. T o b e more preise, the arithmetiit y onlusion of Theorem 5.1 means the follo wing. There exists a global eld K , a onneted semi-simple K -anisotropi K -group H and a nite set Σ of v aluations of K su h that: (i) The quotien t Γ := Γ / Γ ∩ ( R · Q Z ( G )) is ommensurable with the arithmeti group H ( K (Σ)) , where K (Σ) is the ring of Σ -in tegers of K . Moreo v er, Σ on tains all Ar himedean v aluations v for whi h H is K v -isotropi, where K v denotes the v -ompletion of K . In parti- ular, b y BorelHarish-Chandra and BehrHarder redution theory , the diagonal em b edding realises H ( K (Σ)) as a lattie in the pro dut Q v ∈ Σ H ( K v ) . (ii) The group G + is isomorphi to Q v ∈ Σ H ( K v ) + and this isomorphism implemen ts the ommensurabilit y of Γ with H ( K (Σ)) . F or ba kground referenes, inluding on H ( K v ) + , see [ Mar91 , I.3℄. In on trast to statemen ts in [ Mon05 ℄, there is no assumption on the subgroup struture of the fators G i in Theorem 5.1 , whi h ma y not ev en b e irreduible. The nature of the linear represen tation is ho w ev er more restrited. Another impro v emen t is that no (w eak) o ompatness assumption is made on Γ . In partiular, under the same algebrai restritions on the fators G i as in lo . it. , w e obtain the follo wing arithmetiit y vs. non-linearit y alternativ e for all nitely generated latties. Corollary 5.3. L et Γ < G = G 1 × · · · × G n b e an irr e duible nitely gener ate d latti e, wher e e ah G i is a lo al ly omp at gr oup suh that every non-trivial lose d normal sub gr oup is o omp at. Then one of the fol lowing holds: (i) Every nite-dimensional line ar r epr esentation of Γ in har ateristi 6 = 2 , 3 has vir- tual ly soluble image. (ii) G is a semi-simple algebr ai gr oup and Γ is an arithmeti latti e. The h yp othesis made on ea h fator G i ma y b e used to desrib e to some exten t its struture indep enden tly of the existene of a lattie in G ; one an in partiular sho w [ CM08a ℄ that ea h G i is monolithi, th us extending the lassial result of Wilson [ Wil71 ℄ to lo ally ompat groups. Ho w ev er, w e will not app eal to this preliminary desription of the G i when pro ving Corollary 5.3 : the strutural information will instead b e obtained a p osteriori . Remark 5.4. In [ Mon05 ℄, the onlusion ( i ) w as replaed b y niteness of the image. This follo ws from the urren t onlusion in the more restrited setting of lo . it. thanks to Y. Shalom's sup errigidit y for haraters [ Sha00 ℄, unless of ourse G i admits (virtually) a non-zero on tin uous homomorphism to R (after all in the urren t setting w e an ha v e G i = R ). It is part of the assumptions in [ Mon05 ℄ that no su h homomorphism exists, so that Corollary 5.3 indeed generalises lo . it . 26 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD 5.A. Sup errigid pairs. F or on v eniene, w e shall use the follo wing terminology . Let J b e a top ologial group and Λ < J an y subgroup. W e all the pair (Λ , J ) sup errigid if for an y lo al eld k and an y onneted absolutely almost simple adjoin t k -group H , ev ery abstrat homomorphism Λ → H ( k ) with un b ounded Zariski-dense image extends to a on tin uous homomorphism of J . Prop osition 5.5. L et (Λ , J ) b e a sup errigid p air with J lo al ly omp at and Λ nitely gener ate d with losur e of nite ovolume in J . If Λ admits a faithful r epr esentation in an admissible semi-simple gr oup (over any eld) with Zariski-dense image, then the amenable r adi al R of J is omp at and the quasi- entr e Q Z ( J ) is virtual ly ontaine d in Λ · R . F urthermor e, up on r eplaing J by a nite index sub gr oup, the quotient J /R splits as J + × Q Z ( J /R ) wher e J + is a semi-simple algebr ai gr oup and the image of Λ in J + is an arithmeti latti e. (W e p oin t out again that in partiular the diret fator Q Z ( J /R ) is disrete.) One migh t exp et that Theorem 5.1 ould no w b e pro v ed b y establishing in omplete generalit y that nitely generated irreduible latties in pro duts of lo ally ompat groups form a sup errigid pair. F or uniform latties, or more generally w eakly o ompat square- summable latties, this is indeed true and w as pro v ed in [ Mon06 ℄. W e do not ha v e a pro of in general and shall es hew this diult y b y giving rst an indep enden t pro of of the ompatness of the amenable radial (Corollary 5.14 b elo w) and using the residual niteness of nitely generated linear groups b efore pro eeding with Prop osition 5.5 . Nev ertheless, w e do ha v e a general pro of as so on as the groups are totally disonneted. Theorem 5.6. L et Γ < G = G 1 × · · · × G n b e an irr e duible nitely gener ate d latti e, wher e e ah G i is any lo al ly omp at gr oup. If G is total ly dis onne te d, then (Γ , G ) is a sup errigid p air. (As w e shall see in Prop osition 5.11 , one an drop the nite generation assumption in the simpler ase where Γ pro jets faithfully to some fator G i .) The pro ofs will use the follo wing fat established in [ CM08a ℄. Prop osition 5.7. L et G b e a omp atly gener ate d lo al ly omp at gr oup and { N v | v ∈ Σ } b e a ol le tion of p airwise distint lose d normal sub gr oups of G suh that for e ah v ∈ Σ , the quotient H v = G/ N v is quasi-simple, non-disr ete and non- omp at. If T v ∈ Σ N v = 1 then Σ is nite and G has a har ateristi lose d o omp at sub gr oup whih splits as a nite dir e t pr o dut of | Σ | top olo gi al ly simple gr oups. W e reall for the ab o v e statemen t that a group is alled quasi-simple if it p ossesses a o ompat normal subgroup whi h is top ologially simple and on tained in ev ery non-trivial losed normal subgroup. Pr o of of Pr op osition 5.5 . W e will largely follo w the ideas of Margulis, deduing arithmeti- it y from sup errigidit y [ Mar91 , Chapter IX℄. It is assumed that the reader has a op y of [ Mon05 ℄ at hand, sine it on tains a similar reasoning under dieren t h yp otheses. The harateristi assumption in lo . it. will b e replaed b y the urren t admissibilit y assump- tion. The group J (and hene also all nite index subgroups and fators) is ompatly generated b y Lemma 2.12 . Let τ : Λ → H b e the giv en faithful represen tation. Up on replaing Λ and J b y nite index subgroups and p ost-omp osing τ with the pro jetion map H → H / Z ( H ) , w e shall assume heneforth that H is adjoin t and Zariski-onneted. The represen tation ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 27 τ : Λ → H need no longer b e faithful, but it still has nite k ernel. As in [ Mon05 , (3.3)℄, in view of the assumption that Λ is nitely generated, w e ma y assume that H is dened o v er a nitely generated eld K . This is the rst of t w o plaes where the admissibilit y assumption is used in lo . it. (follo wing VI I I.3.22 and IX.1.8 in [ Mar91 ℄). By Tits' alternativ e [ Tit72 ℄, the amenable radial of Λ is soluble-b y-lo ally-nite and th us lo ally nite sine τ (Λ) is Zariski-dense and H is semi-simple. The nite generation of K implies that this radial is in fat nite (see e.g. Corollary 4.8 in [ W eh73 ℄), th us trivial b y Zariski-densit y sine H is adjoin t. (The nite generation of K is essen tial in p ositiv e harateristi sine for algebraially losed elds there is alw a ys a lo ally nite Zariski-dense subgroup [ BGM04 ℄.) It no w follo ws that if J is a ompat extension of a disrete group, then the latter has trivial amenable radial and th us all the onlusions of Prop osition 5.5 hold trivially . Therefore, w e assume heneforw ard that J is not ompat-b y-disrete. Let H = H 1 × · · · × H k b e the deomp osition of H in to its simple fators. W e shall w ork with the fators H i one at a time. Let τ i : Λ → H i b e the indued represen tation of Λ . Notie that τ i need not b e faithful; ho w ev er, it has Zariski-dense (and in partiular innite) image. W e let Σ i denote the set of all (inequiv alen t represen tativ es of ) v aluations v of K su h that the image of τ i (Λ) is not relativ ely ompat in H i ( K v ) (for the Hausdor top ology); observ e that this image is still Zariski-dense. Then Σ i is non-empt y sine τ i (Λ) is innite, see [ BG07 , Lemma 2.1℄. By h yp othesis, there exists a on tin uous represen tation J → H i ( K v ) for ea h v ∈ Σ i , extending the giv en Λ -represen tation. W e denote b y N v ✁ J the k ernel of this represen tation. Let I ⊆ { 1 , . . . , k } b e the set of all those indies i su h that J / N v is non-disrete for ea h v ∈ Σ i . W e laim that the set I is non-empty. Indeed, for ea h index j 6∈ I , there exists v j ∈ Σ j su h that N v j is op en in J . Th us the k ernel J + = \ j 6∈ I N v j of the on tin uous represen tation J → Q j 6∈ I H j ( K v j ) is op en. By assumption the losure of Λ in J has nite o v olume. Therefore, for ea h op en subgroup F < J , the losure of Λ ∩ F has nite o v olume in F . It follo ws in partiular that Λ ∩ F is innite unless F is ompat. These onsiderations apply to the op en subgroup J + < J . Sine J is not ompat-b y- disrete, w e dedue that Λ ∩ J + is innite. Therefore the restrition to Λ of the represen tation J → Q j 6∈ I H j ( K v j ) has innite k ernel and, hene, it do es not fator through τ : Λ → H ( K ) . In partiular it annot oinide with the giv en represen tation τ : Λ → H . Th us I is non- empt y . W e laim that for e ah i ∈ I , the set Σ i is nite. Let i ∈ I and v ∈ Σ i . The argumen ts of [ Mon05 , (3.7)℄ sho w that the isomorphi image of J / N v in H i ( K v ) on tains H i ( K v ) + . These argumen ts use again the admissibilit y assumption b eause the app eal to a result of R. Pink [ Pin98 ℄; the fat that the latter hold in the admissible ase is expliit in the table pro vided in Prop osition 1.6 of [ Pin98 ℄. F urthermore, it follo ws from Tits' simpliit y theorem [ Tit64 ℄ om bined with [ BT73 , 6.14℄ that ea h J / N v is quasi-simple. Moreo v er, an appliation of [ BT73 , 8.13℄ sho ws that the v arious quotien ts 28 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD ( J / N v ) v ∈ Σ i are pairwise non-isomorphi. In partiular the normal subgroups ( N v ) v ∈ Σ i are pairwise distint. Let D i = T v ∈ Σ i N v and reall that J /D i is ompatly generated. Pro jeting ea h N v to J /D i , w e obtain a family of pairwise distint normal subgroups of J /D i indexed b y Σ i su h that ea h orresp onding quotien t is quasi-simple, non-disrete and non-ompat. Therefore, the desired laim follo ws from Prop osition 5.7 . In partiular, app ealing again to [ BT73 , Corollaire 8.13℄, w e obtain a on tin uous map J → Q v ∈ Σ i H i ( K v ) whi h w e denote again b y τ J i . The k ernel of τ J i is D i . Up on replaing J and Λ b y nite index subgroups w e ma y assume that the image of τ J i oinides in fat with Q v ∈ Σ i H i ( K v ) + , ompare [ Mon05 , (3.9)℄. W e laim that R := J + ∩ D is omp at and that J = J + · D , wher e D is dene d by D = T i ∈ I D i . W e rst sho w that R = J + ∩ D is ompat. Assume for a on tradition that this is not the ase. Then, giv en a ompat op en subgroup U of J , the in tersetion Λ 0 = Λ ∩ ( U · R ) is innite: this follo ws from the same argumen t as ab o v e, using the assumption that the losure of Λ has nite o v olume. F or ea h index j 6∈ I , w e ha v e J + ⊆ N v j and w e dedue that the image of Λ 0 in H j ( K v j ) is nite, sine it is on tained in the image of U . Equiv alen tly , the subgroup τ j (Λ 0 ) < H j ( K ) is nite. It follo ws in partiular that τ i (Λ 0 ) is innite for some i ∈ I . By [ BG07 , Lemma 2.1℄, there exists v ∈ Σ i su h that the image of Λ 0 in H i ( K v ) is un b ounded. This is absurd sine D ⊆ N v and hene the image of Λ 0 in H i ( K v ) is on tained in the image of the ompat subgroup U . This sho ws that the in tersetion R is indeed ompat. A t this p oin t w e kno w that the quotien t J /D is isomorphi to a subgroup of the pro dut Y i ∈ I Y v ∈ Σ i H i ( K v ) + whi h pro jets surjetiv ely on to ea h fator of the form Q v ∈ Σ i H i ( K v ) + . Using again the Goursat-t yp e argumen t as in Prop osition 5.7 , w e nd that J /D is indeed isomorphi to a nite pro dut of non-ompat non-disrete simple groups H i ( K v ) + . In partiular the quotien t J /D has no non-trivial op en normal subgroup. Sine J + is op en and normal in J , w e dedue that J = J + · D , thereb y establishing the laim. By the v ery nature of the statemen t, w e ma y replae J b y the quotien t J /R without an y loss of generalit y , sine R is ompat. In view of this further simpliation, the preeding laim implies that J ∼ = J + × D . In partiular D is disrete. It no w follo ws as in [ Mon05 , (3.11)℄ that K is a global eld, and that the image of Λ in the semi-simple group J /D is an arithmeti lattie (ompare [ Mon05 , (3.13)℄). Therefore, b y Prop osition 3.1 , the in tersetion Λ ∩ D is a lattie in D and, hene, the disrete normal subgroup D is virtually on tained in Λ . As J ∼ = J + × D and J + has trivial quasi-en tre, it follo ws that the quasi-en tre of J oinides with D . This nishes the pro of. F or later use, w e single out a (simpler) v ersion of an argumen t referred to ab o v e. Lemma 5.8. L et H b e an admissible onne te d absolutely almost simple adjoint k -gr oup H , wher e k is a lo al eld. L et J b e a lo al ly omp at gr oup with a ontinuous unb ounde d Zariski-dense homomorphism τ : J → H ( k ) . Then any omp at normal sub gr oup of J is ontaine d in the kernel of τ . ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 29 Pr o of. Let K ✁ J b e a ompat normal subgroup. The Zariski losure of τ ( K ) is normalised b y the Zariski-dense group τ ( J ) and therefore it is either H ( k ) or trivial. W e assume the former sine otherwise w e are done. W e laim that w e an assume k non-Ar himedean. Otherwise, either k = R or k = C . In the rst ase, τ ( K ) oinides with its Zariski losure b y W eyl's algebraiit y theorem [ Vin94 , 4.2.1℄ so that H ( k ) is ompat in whi h ase the lemma is v oid b y the un b oundedness assumption. In the seond ase, one an redue to the ase τ ( J ) ⊆ H ( R ) as in [ Mon05 , (3.5)℄ and th us τ ( K ) = 1 as b efore sine H ( R ) is also simple; the laim is pro v ed. F ollo wing no w an idea from [ Sha00 , p. 41℄ (see also the explanations in Setion (3.7) of [ Mon05 ℄), one uses [ Pin98 ℄ to dedue that τ ( K ) is op en up on p ossibly replaing k b y a losed subeld (the admissibilit y assumption en ters as in the pro of of Prop osition 5.5 ). W e an still denote this subeld b y k b eause it aommo dates the whole image τ ( J ) , see again [ Mon05 , (3.7)℄. No w τ ( J ) is an un b ounded op en subgroup and hene on tains H ( k ) + b y a result of J. Tits (see [ Pra82 ℄; this also follo ws from the Ho w eMo ore theorem [ HM79 ℄ whi h ho w ev er is p osterior to Tits' result). This implies that the ompat group τ ( K ) is trivial sine H ( k ) + is simple b y [ Tit64 ℄. 5.B. Boundary maps. W e reord t w o statemen ts extrated from Margulis' w ork in the form most on v enien t for us. Prop osition 5.9. L et J b e a se ond ountable lo al ly omp at gr oup with a me asur e lass pr eserving ation on a standar d pr ob ability sp a e B . L et Λ < J b e a dense sub gr oup with a Zariski-dense unb ounde d r epr esentation τ : Γ → H ( k ) to a onne te d absolutely almost k -simple adjoint gr oup H over an arbitr ary lo al eld k . If ther e is a pr op er k -sub gr oup L < H and a Λ -e quivariant non-essential ly- onstant me a- sur able map B → H ( k ) / L ( k ) , then τ extends to a ontinuous homomorphism J → H ( k ) . Pr o of. The argumen t is giv en b y A'Camp oBurger in the harateristi zero ase at the end of Setion 7 in [ AB94 ℄ (pp. 1819). This referene onsiders homogeneous spaes for B but this restrition is nev er used. The general statemen t is referred to in [ Bur95 ℄ and details are giv en in [ Bon04 ℄. Prop osition 5.10. L et Γ b e a ountable gr oup with a Zariski-dense unb ounde d r epr esenta- tion Γ → H ( k ) to a onne te d absolutely almost k -simple adjoint gr oup H over an arbitr ary lo al eld k . L et B b e a standar d pr ob ability sp a e with a me asur e lass pr eserving Γ -ation that is amenable in Zimmer's sense [ Zim84 ℄ and suh that the diagonal ation on B 2 is er go di. Then ther e is a pr op er k -sub gr oup L < H and a Γ -e quivariant non-essential ly- onstant me asur able map B → H ( k ) / L ( k ) . Pr o of. Again, this is pro v ed in [ AB94 ℄ for the harateristi zero ase (and B homogeneous) and the neessary adaptations to the general ase are explained in [ Bon04 ℄. W e shall need these sp ei statemen ts b elo w. They rst app eared within the pro of of Margulis' ommensurator sup errigidit y , whi h an adapted as follo ws using [ Bur95 ℄ and Lemma 8.3 in [ CM08 ℄, pro viding a rst step to w ards Theorem 5.6 . Prop osition 5.11. L et G = G 1 × G 2 b e a pr o dut of lo al ly omp at σ - omp at gr oups and Λ < G b e an irr e duible latti e. Assume that the pr oje tion of Λ to G 1 is inje tive and that G 2 admits a omp at op en sub gr oup. Then the p air (Λ , G ) is sup errigid. Pr o of. W e laim that one an assume G seond oun table. As explained in [ Mon06 , Prop osi- tion 61℄, σ -ompatness implies the existene of a ompat normal subgroup K ✁ G meeting 30 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Λ trivially and su h that G/K is seond oun table. Applying the statemen t to G/K to- gether with the image of Λ therein yields the general statemen t sine the pro jetion of Λ to G/K is an isomorphism; this pro v es the laim. Let τ : Λ → H ( k ) b e as in the denition of sup errigid pairs and let U < G 2 b e a ompat op en subgroup. Set Λ U = Λ ∩ ( G 1 × U ) . By the injetivit y assumption and Lemma 3.2 , w e an onsider Λ U as a lattie in G 1 whi h is ommensurated b y (the image of the pro jetion of ) Λ . W e distinguish t w o ases. Assume rst that τ (Λ U ) is un b ounded in the lo ally ompat group H ( k ) . W e ma y then apply Margulis' ommensurator sup errigidit y in its general form prop osed b y M. Burger [ Bur95 , Theorem 2.A℄, see [ Bon04 ℄ for details. This yields a on tin uous map J → H ( k ) fatoring through G 1 and extending the giv en Λ -represen tation, as desired. Assume no w that τ (Λ U ) is b ounded, whi h is equiv alen t to Λ U xing a p oin t in the symmetri spae or BruhatTits building asso iated to H ( k ) . Then Lemma 8.3 in [ CM08 ℄ yields a on tin uous map J → H ( k ) fatoring through G 2 . 5.C. Radial sup errigidit y. Theorem 5.12. L et G b e a lo al ly omp at gr oup, R ✁ G its amenable r adi al, Γ < G a nitely gener ate d latti e and F the losur e of the image of Γ in G/R . Then any Zariski-dense unb ounde d r epr esentation of Γ in any onne te d absolutely almost simple adjoint k -gr oup H over any lo al eld k arises fr om a ontinuous r epr esentation of F via the map Γ → F . (In partiular, the pair (Γ / (Γ ∩ R ) , F ) is sup errigid.) Pr o of. Notie that G is σ -ompat sine it on tains a nitely generated, hene oun table, lattie. (In fat G is ev en ompatly generated b y Lemma 2.12 .) Set J = G/R . There exists a standard probabilit y J -spae B on whi h the Γ -ation is amenable and su h that the diagonal Γ -ation on B 2 is ergo di; it sues to ho ose B to b e the P oisson b oundary of a symmetri random w alk with full supp ort on J . Indeed: (i) The J -ation is amenable as w as sho wn b y Zimmer [ Zim78 ℄; this implies that the G -ation is amenable sine R is an amenable group and th us that the Γ -ation is amenable sine Γ is losed in G (see [ Zim84 , 5.3.5℄). (ii) The diagonal ation of an y losed nite o v olume subgroup F < J on B 2 is ergo di in view of the er go diity with o eients of J , and hene the same holds for dense subgroups of F . F or detailed ba kground on this strengthening of ergo diit y in tro dued in [ BM02 ℄ and on the P oisson b oundary in general, w e refer the reader to [ Ka 03 ℄. Let no w k b e a lo al eld, H a onneted absolutely almost simple k -group and Γ → H ( k ) a Zariski-dense un b ounded represen tation. W e an apply Prop osition 5.10 and obtain a prop er subgroup L < H and a Γ -equiv arian t map B → H ( k ) / L ( k ) . W riting Λ for the image of Γ in J , w e an therefore apply Prop osition 5.9 with F instead of J and the onlusion follo ws. Remark 5.13. An examination of this pro of sho ws that one has also the follo wing related result. Let J b e a seond oun table lo ally ompat group and Λ ⊆ J a dense oun table subgroup whose ation on J b y left m ultipliation is amenable. Then the pair (Λ , J ) is sup errigid. Indeed, one an again argue with Prop ositions 5.9 and 5.10 b eause it is easy to he k that in the presen t situation an y amenable J -spae is also amenable for Λ view ed as a disrete group. Related ideas w ere used b y R. Zimmer in [ Zim87 ℄. Corollary 5.14. L et G b e a lo al ly omp at gr oup and Γ < G a nitely gener ate d latti e. If Γ admits a faithful Zariski-dense r epr esentation in an admissible semi-simple gr oup (over any eld), then the amenable r adi al of G is omp at. ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 31 Pr o of. Let R b e the amenable radial of G , F b e the losure of the image of Γ in G/R and J < G the preimage of F in G . The on ten t of Theorem 5.12 is that the pair (Γ , J ) is sup errigid. Sine in addition Γ is losed and of nite o v olume in J (see [ Rag72 , Lemma 1.6℄), w e ma y apply Prop osition 5.5 and dedue that the amenable radial of J is ompat. The onlusion follo ws sine R < J . 5.D. Latties with non-disrete ommensurators. The follo wing useful tri k allo ws to realise the ommensurator of an y lattie in a lo ally ompat group G as a lattie in a pro dut G × D . A similar reasoning in the sp eial ase of automorphism groups of trees ma y b e found in [ BG02 , Theorem 6.6℄. Lemma 5.15. L et Λ b e a gr oup and Γ < Λ a sub gr oup ommensur ate d by Λ . L et D b e the ompletion of Λ with r esp e t to the left or right uniform strutur e gener ate d by the Λ - onjugates of Γ . Then D is a total ly dis onne te d lo al ly omp at gr oup. If furthermor e G is a lo al ly omp at gr oup ontaining Λ as a dense sub gr oup suh that Γ is disr ete (r esp. is a latti e) in G , then the diagonal emb e dding of Λ in G × D is disr ete (r esp. is an irr e duible latti e). The ab o v e lemma is in some sense a on v erse to Lemma 3.2 . In the sp eial ase where one starts with a lattie satisfying a faithfulness ondition, this relation b eomes ev en stronger. Lemma 5.16. L et G, H b e lo al ly omp at gr oups and Λ < G × H a latti e. Assume that the pr oje tion of Λ to G is faithful and that b oth pr oje tions ar e dense. L et U < H b e a omp at op en sub gr oup, set Γ = Λ ∩ ( G × U ) as in L emma 3.2 and onsider the gr oup D as in L emma 5.15 (up on viewing Λ as a sub gr oup of G ). Dene the omp at normal sub gr oup K ✁ H as the or e K = T h ∈ H hU h − 1 of U in G . Then the map Λ → D indu es an isomorphism of top olo gi al gr oups H /K ∼ = D . Pr o of of L emma 5.15 . One v eries readily the ondition giv en in [ Bou60 ℄ (TG I I I, 3, No 4, Théorème 1) ensuring that the ompletion satises the axioms of a group top ology . W e emphasise that it is part of the denition of the ompletion that D is Hausdor; in other w ords D is obtained b y rst ompleting Λ with resp et to the group top ology as dened ab o v e, and then dividing out the normal subgroup onsisting of those elemen ts whi h are not separated from the iden tit y . Let U denote the losure of the pro jetion of Γ to D . By denition U is op en. Notie that it is ompat sine it is a quotien t of the pronite ompletion of Γ b y onstrution. In partiular D is lo ally ompat. By a sligh t abuse of notation, let us iden tify Γ and Λ with their images in D . W e laim that U ∩ Λ = Γ . Indeed, let { γ n } n ≥ 0 b e a sequene of elemen ts of Γ su h that lim n γ n = λ ∈ Λ . Sine λ Γ λ − 1 is a neigh b ourho o d of the iden tit y in Λ (with resp et to the top ology indued from D ), it follo ws that γ n λ − 1 ∈ λ Γ λ − 1 for n large enough. Th us λ ∈ γ n Γ = Γ . Assume no w that Γ is disrete and ho ose a neigh b ourho o d V of the iden tit y in G su h that Γ ∩ V = 1 . In view of the preeding laim the pro dut V × U is a neigh b ourho o d of the iden tit y in G × D whi h meets Λ trivially , thereb y sho wing that Λ is disrete. Assume nally that Γ is a lattie in G and let F b e a fundamen tal domain. Then F × U is a fundamen tal domain for Λ in G × D , whi h has nite v olume sine a Haar measure for G × D ma y b e obtained b y taking the pro dut of resp etiv e Haar measures for G and D . Th us Λ has nite o v olume in G × D . Pr o of of L emma 5.16 . In order to onstrut a on tin uous homomorphism π : H → D , it sues to he k that an y net in Λ whose image in H on v erges to the iden tit y also on v erges 32 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD to the iden tit y in D ; this follo ws from the denitions of Γ and D sine the net is ev en tually in an y Λ -onjugate of U . Notie that π has dense image. W e laim that the k ernel of π is T λ ∈ Λ λU λ − 1 . Indeed, if on the one hand k ∈ ke r( π ) is the limit of the images in H of a net { λ i } in Λ , then for an y λ w e ha v e ev en tually λ i ∈ λ − 1 Γ λ ⊆ λ − 1 ( G × U ) λ so that indeed k ∈ λ − 1 U λ sine U is losed. Con v ersely , if k ∈ T λ ∈ Λ λU λ − 1 is limit of images of { λ i } , then, sine U is op en, for an y λ the image of λ i is ev en tually in λU λ − 1 , hene in λ Γ λ − 1 so that π ( λ i ) → 1 . This pro v es the laim. No w it follo ws that k er( π ) is indeed the ore K of the statemen t sine U is ompat. The fat that π is on to and op en follo ws from the existene of a ompat op en subgroup in H . Theorem 5.17. L et G b e a lo al ly omp at gr oup and Γ < G b e a latti e. Assume that G p ossesses a nitely gener ate d dense sub gr oup Λ suh that Γ < Λ < Comm G (Γ) . If Λ admits a faithful Zariski-dense r epr esentation in an admissible semi-simple gr oup (over any eld), then the amenable r adi al R of G is omp at and the quasi- entr e Q Z ( G ) is virtual ly ontaine d in Γ · R . F urthermor e, up on r eplaing G by a nite index sub gr oup, the quotient G/R splits as G + × Q Z ( G/R ) wher e G + is a semi-simple algebr ai gr oup and the image of Γ in G + is an arithmeti latti e. Pr o of. Let J = G × D , where D is the totally disonneted lo ally ompat group pro vided b y Lemma 5.15 . As a totally disonneted group, it has n umerous ompat op en subgroups (for instane the losure of Γ ). W e shall view Λ as an irreduible lattie in J . The pro jetion of Λ to G is faithful b y onstrution. By Prop osition 5.11 , the pair (Λ , J ) is sup errigid. This allo ws us to apply Prop osition 5.5 . Sine the amenable radial R G of G is on tained in the amenable radial R J of J , it is ompat. F urthermore, the quasi-en tre of G is on tained in the quasi-en tre of J and the en tre-free group G/R G is a diret fator of J + × Q Z ( J /R J ) ; the desired onlusions follo w. 5.E. Latties in pro duts of Lie and totally disonneted groups. Theorem 5.18. L et Γ < G = S × D b e a nitely gener ate d irr e duible latti e, wher e S is a onne te d semi-simple Lie gr oup with trivial entr e and D is a total ly dis onne te d lo al ly omp at gr oup. L et Γ D ✁ D b e the anoni al disr ete kernel of D . Then D / Γ D is a pr onite extension of a semi-simple algebr ai gr oup Q and the image of Γ in S × Q , whih is isomorphi to Γ / Γ D , is an arithmeti latti e. Corollary 5.19. In p artiular, D is lo al ly pr onite by analyti. A family of examples will b e onstruted in Setion 6.C b elo w, sho wing that the statemen t annot b e simplied ev en in a geometri setting (see Remark 6.7 ). Pr o of of The or em 5.18 . By the v ery nature of the statemen t, w e an fator out the anonial disrete k ernel. Therefore, w e shall assume heneforth that the pro jetion map Γ → S is injetiv e. W e an also assume that S has no ompat fators. Sine S is onneted with trivial en tre, there is a Zariski onneted semi-simple adjoin t R -group H without R - anisotropi fators su h that S = H ( R ) . Notie that the injetivit y of Γ → S is preserv ed when passing to nite index subgroups. By Prop osition 5.11 , the pair (Γ , G ) is sup errigid. W e an therefore apply Prop osition 5.5 . In partiular, D has ompat amenable radial and therefore, in view of the statemen t of Theorem 5.18 , w e an assume that this radial is trivial. Giv en the onlusion of Prop osi- tion 5.5 , it only remains to sho w that the quasi-en tre Q Z ( G ) of G is trivial. W e no w kno w that Q Z ( G ) is virtually on tained in Γ ; sine on the other hand S has trivial quasi-en tre, ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 33 Q Z ( G ) ⊆ 1 × D . In other w ords, Q Z ( G ) is on tained in the disrete k ernel Γ D , whi h has b een rendered trivial. This ompletes the pro of. W e ha v e treated Theorem 5.18 as a p ort of all on the w a y to Theorem 5.1 . In fat, one an also desrib e latties in pro duts of groups with a simple algebrai fator o v er an arbitrary lo al eld and in most ases without assuming nite generation a priori . W e reord the follo wing statemen t, whi h will not b e used b elo w. Theorem 5.20. L et k b e any lo al eld and G an admissible onne te d absolutely almost simple adjoint k -gr oup. L et H b e any omp atly gener ate d lo al ly omp at gr oup admitting a omp at op en sub gr oup. L et Γ < G ( k ) × H b e an irr e duible latti e. In ase k has p ositive har ateristi and the k -r ank of G is one, we assume Γ o omp at. Then H / Γ H is a omp at extension of a semi-simple algebr ai gr oup Q and the image of Γ in G ( k ) × Q is an arithmeti latti e. There is no assumption whatso ev er on the ompatly generated lo ally ompat group H b ey ond admitting a ompat op en subgroup; reall that the latter is automati if H is totally disonneted [ Bou71 , I I I 4 No 6℄. Notie that a p osteriori it follo ws from arithmetiit y that Γ is nitely generated; in the pro of b elo w, nite generation will b e established in t w o steps. Pr o of of The or em 5.20 . W e fator out the anonial disrete k ernel Γ H and assume hene- forth that it is trivial. This do es not aet the other assumptions and th us w e ho ose some ompat op en subgroup U < H . W e write G = G ( k ) and onsider Γ U = Γ ∩ ( G × U ) as in Lemma 3.2 . Sine w e fatored out the anonial disrete k ernel, w e an onsider Γ U as a lattie in G ommensurated b y the dense subgroup Γ < G . Moreo v er, Γ U is nitely generated; indeed, either w e ha v e sim ultaneously rank k ( G ) = 1 and c har( k ) > 0 , in whi h ase w e assumed Γ o ompat, so that Γ U is o ompat in the ompatly generated group G ( k ) (again Lemma 3.2 ) and hene nitely generated [ Mar91 , I.0.40℄; or else, Γ U is kno wn to b e nitely generated b y applying, as the ase ma y b e, either Kazhdan's prop ert y , or the theory of fundamen tal domains, or the o ompatness of p -adi latties w e refer to Margulis, Setions (3.1) and (3.2) of Chapter IX in [ Mar91 ℄. W e an no w apply Margulis' arthmetiit y [ Mar91 , 1.(1)℄ and dedue that G is dened o v er a global eld K and that Γ U is ommensurable to G ( K ( S )) for some nite set of plaes S ; in short Γ U is S -arithmeti. (The idea to obtain rst this preliminary arithmetiit y of Γ U w as suggested b y M. Burger.) It follo ws that Γ is rational o v er the global eld K , see Theorem 3.b in [ Bor66 ℄ ( lo . it. is form ulated for the Lie group ase; see [ W or07 , Lemma 7.3℄ in general). Sine the pair (Γ , G × H ) is sup errigid (for instane b y Prop osition 5.11 ), only the a priori la k of nite generation for Γ prev en ts us from applying Prop osition 5.5 . Ho w ev er, a go o d part of the pro of of that prop osition is already seured here sine Γ has b een sho wn to b e rational o v er a global eld. W e no w pro eed to explain ho w to adapt the remaining part of that pro of to the urren t setting. W e use those elemen ts of notation in tro dued in the pro of of Prop osition 5.5 that do not onit with presen t notation and review all uses of nite generation that are either expliit in the pro of of Prop osition 5.5 or impliit through referenes to [ Mon05 ℄. The ompat generation of G × H is an assumption rather than a onsequene of Lemma 2.12 . W e also used nite generation in order to pass to a nite index subgroup of Γ on tained in G ( K v ) + for all v aluations v ∈ Σ . W e shall p ostp one this step, so that the whole argu- men tation pro vides us with maps from G × H to a pro dut Q of fators that lie in-b et w een 34 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD G ( K v ) + and G ( K v ) . In partiular all these fators are quasi-simple and w e an still app eal to Prop osition 5.7 as b efore. Notie ho w ev er that at the v ery end of the pro of, one nite generation is gran ted, w e an in v ok e the argumen t that G ( K v ) / G ( K v ) + is virtually torsion Ab elian [ BT73 , 6.14℄ and th us redue again to the ase where Γ is on tained in G ( K v ) + . W e no w justify that the image of Γ in G × Q is disrete b eause previously this follo w ed from [ Mon05 , (3.13)℄ whi h relies on nite generation. If Γ w ere not disrete, an appliation of [ BG07 , Lemma 2.1℄ w ould pro vide a v aluation v / ∈ Σ with Γ un b ounded in G ( K v ) , whi h is absurd. W e are no w in a situation where G × H maps to G × Q with o ompat nite o v olume image and injetiv ely on Γ ; therefore the disreteness of the image of Γ implies that this map is prop er and hene H is a ompat extension of Q . Pushing forw ard the measure on ( G × H ) / Γ , w e see that the image of Γ in G × Q is a lattie. No w Γ is nitely generated (see ab o v e referenes to [ Mar91 , IX℄) and th us the pro of is ompleted as in Prop osition 5.5 . The disrete fator o urring in the onlusion of the latter prop osition is trivial for the same reason as in the pro of of Theorem 5.18 . 5.F. Latties in general pro duts. W e b egin with the sp eial ase of totally disonneted groups. Pr o of of The or em 5.6 . An issue that w e need to deal with is that the pro jetion of Γ to G 1 is a priori not faithful. In order to irum v en t this diult y , w e pro eed to a preliminary onstrution. Let ι : Γ → b Γ b e the anonial map to the pronite ompletion of Γ and denote its k ernel b y Γ (f ) ; in other w ords, Γ (f ) is the nite residual of Γ . Let b G 1 denote the lo ally ompat group whi h is dened as the losure of the image of Γ in G 1 × b Γ under the pro dut map pro j 1 × ι , where pro j 1 : G → G 1 is the anonial pro jetion. Sine pro j 1 (Γ) is dense in G 1 and b Γ is ompat, the anonial map b G 1 → G 1 is surjetiv e. In other w ords, the group b G 1 is a ompat extension of G 1 . W e no w dene G ′ 1 = G 2 × · · · × G n and b G = b G 1 × G ′ 1 . Then Γ admits a diagonal em b edding in to b G through whi h the injetion of Γ in G fators. W e will heneforth iden tify Γ with its image in b G and onsider Γ as an irreduible lattie of b G . W e laim that the p air (Γ , b G ) is sup errigid . The argumen t is a v ariation on the pro of of Prop osition 5.11 . Let τ : Γ → H ( k ) b e as in the denition of sup errigid pairs. Sine τ (Γ) is nitely generated and linear, it is residually nite [ Mal40 ℄. This means that τ fators through Γ := Γ / Γ (f ) . Let U < G ′ 1 b e a ompat op en subgroup, Γ U = Γ ∩ ( b G 1 × U ) and Γ U = Γ U / (Γ U ∩ Γ (f ) ) . By onstrution and Lemma 3.2 , w e an onsider Γ U as a lattie in b G 1 ommensurated b y Γ . Arguing as in Prop osition 5.11 , when τ (Γ U ) is un b ounded one applies ommensurator sup errigidit y yielding a on tin uous map J → H ( k ) and extending the map Γ → H ( k ) and hene also τ . When τ (Γ U ) is b ounded, one applies Lemma 8.3 in [ CM08 ℄ instead and the resulting extension fators through G ′ 1 . This pro v es the laim. In order to onlude that the pair (Γ , G ) is also sup errigid, it no w sues to apply Lemma 5.8 . Corollary 5.21. The or em 5.1 holds in the p artiular ase of total ly dis onne te d gr oups. Pr o of. Theorem 5.6 pro vides the h yp othesis needed for Prop osition 5.5 . W e no w turn to the general ase Γ < G = G 1 × · · · × G n of Theorem 5.1 . The main part of the remaining pro of will onsist of a areful analysis of ho w the lattie Γ migh t sit ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 35 in v arious subpro duts hidden in the fators G i or their nite index subgroups one the amenable radial has b een trivialised. It will turn out that Γ is virtually a diret pro dut Γ ′ × Γ ′′ , where Γ ′ is an irreduible lattie in a pro dut S ′ × D ′ with S ′ a semi-simple Lie (virtual) subpro dut of G and D ′ a totally disonneted subgroup of G whose p osition will b e laried; as for Γ ′′ , it is an irreduible lattie in a semi-simple Lie group S ′′ that turns out to satisfy the assumptions of Margulis' arithmetiit y . Of ourse, an y of the ab o v e fators migh t w ell b e trivial. Completion of the pr o of of The or em 5.1 . The amenable radial is ompat b y Corollary 5.14 and hene w e an assume that it is trivial. The group G (and hene also all nite index subgroups and fators) is ompatly generated b y Lemma 2.12 . Up on regrouping the last n − 1 fators and in view of the denition of an irreduible lattie (see p. 18 ), w e an assume G = G 1 × G 2 . W e apply the solution to Hilb ert's fth problem (ompare Theorem 5.6 in [ CM08 ℄) and write G i = S i × D i after replaing G and Γ with nite index subgroups. Here S i are onneted semi-simple en tre-free Lie groups without ompat fators and D i totally disonneted ompatly generated with trivial amenable radial. Set S = S 1 × S 2 and D = D 1 × D 2 . Th us Γ is a lattie in G = S × D . Notie that if S is trivial, then G is totally disonneted and w e are done b y Theorem 5.6 . W e assume heneforth that S is non-trivial. The main remaining obstale is that the lattie Γ need not b e irreduible with resp et to the pro dut deomp osition G = S × D . Observe that the losur e pro j D (Γ) of the pr oje tion of Γ to D has trivial amenable r adi al. Indeed pro j D i (Γ) is dense in Γ i for i = 1 , 2 , hene the pro jetion pro j D (Γ) → D i has dense image. The desired laim follo ws sine G , and hene D i , has trivial amenable radial. Let U < D b e a ompat op en subgroup and set Γ U = Γ ∩ ( S × U ) . By Lemma 3.2 , the pro jetion pro j S (Γ U ) of Γ U to S is a lattie whi h is ommensurated b y pro j S (Γ) . The lattie pro j S (Γ U ) p ossesses a nite index subgroup whi h admits a anonial splitting in to nitely man y irreduible groups Γ 1 × · · · × Γ r , ompare Theorem 4.2 . F urthermore ea h Γ i is an irreduible lattie in a semi-simple subgroup S i < S whi h is obtained b y regrouping some of the simple fators of S . Sine the pro jetion of Γ to ea h G 1 and G 2 , and hene to S 1 and S 2 , is dense, it follo ws that the pro jetion of Γ to ea h simple fator of S is dense. W e no w onsider the pro jetion of Γ to the v arious fators S i . In view of the preeding remark and the fat that Γ i is an irreduible lattie in S i , it follo ws that pro j S i (Γ) is either dense in S i or disrete and on tains Γ i with nite index, see [ Mar91 , IX.2.7℄. Let no w S ′ = h S i | pro j S i (Γ) is non-disrete i and S ′′ = h S i | pro j S i (Γ) is disrete i . W e laim that the pr oje tion of Γ to S ′ is dense. If this failed, then b y [ Mar91 , IX.2.7℄ there w ould b e a subpro dut of some simple fators of S ′ on whi h the pro jetion of Γ is a lattie. Sine ea h Γ i is irreduible, this subpro dut is a regrouping S i 1 × · · · × S i p of some fators S i . No w the pro jetion of Γ is a lattie in this subgroup, hene it on tains the pro dut Γ i 1 × · · · × Γ i p with nite index and th us pro jets disretely to ea h S i j . This on tradits the denition of S ′ and pro v es the laim. Our next laim is that Γ has a nite index sub gr oup whih splits as Γ ′ × Γ ′′ , wher e Γ ′′ = pro j S ′′ (Γ) and Γ ′ is a latti e in S ′ × D . 36 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD In order to establish this, w e dene Γ ′ = Ker(pro j : Γ → S ′′ ) and Γ ′′ = \ γ ∈ Γ γ Γ U γ − 1 . Notie that Γ ′ and Γ ′′ are b oth normal subgroups of Γ . Sine pro j D (Γ ′′ ) is a ompat subgroup of D normalised b y pro j D (Γ) , whi h has trivial amenable radial, it follo ws that Γ ′′ ⊂ S ′ × S ′′ × 1 . Therefore, the in tersetion Γ ′ ∩ Γ ′′ is a normal subgroup of Γ on tained in S ′ × 1 × 1 . In view of the preeding laim, w e dedue that Γ ′ ∩ Γ ′′ = 1 . Th us h Γ ′ ∪ Γ ′′ i ✁ Γ is isomorphi to Γ ′ × Γ ′′ . Sine Γ ′ U = Γ ′ ∩ Γ U pro jets to a lattie in S ′ whi h omm utes with the pro jetion of Γ ′′ , w e dedue moreo v er that pro j S ′ (Γ ′′ ) = 1 , or equiv alen tly that Γ ′′ < 1 × S ′′ × 1 . Sine the pro jetion of Γ to S ′′ has disrete image b y denition, it follo ws from Prop o- sition 3.1 that Γ ′ < S ′ × 1 × D pro jets on to a lattie in S ′ × D . On the other hand, the v ery denition of S ′′ implies pro j S ′′ (Γ) on tains pro j S ′′ (Γ U ) , and hene also pro j S ′′ (Γ ′′ ) , as a nite index subgroup. In partiular, this sho ws that Γ ′ × Γ ′′ is a lattie in S ′ × S ′′ × D . Sine it is on tained in the lattie Γ , w e nally dedue that the index of Γ ′ × Γ ′′ in Γ is nite. W e observ e that w e ha v e in partiular obtained a lattie Γ ′′ < S ′′ with S ′′ non-simple and Γ ′′ irreduible (unless b oth Γ ′′ and S ′′ are trivial), b eause the pro jetion of Γ to an y simple Lie group fator is dense: indeed, an y simple fator m ust b e a fator of some G i and Γ pro jets densely on G i . It follo ws from Margulis' arithmetiit y theorem [ Mar91 , Theorem 1.(1')℄ that Γ ′′ is an arithmeti lattie in S ′′ . T urning to the other lattie, w e remark that Γ ′ admits a faithful Zariski-dense represen- tation in a semi-simple group, obtained b y reduing the giv en represen tation of Γ . F urther- more, notie that the pro jetion of Γ to S ′ oinides (virtually) with the pro jetion of Γ . In partiular it has dense image. Therefore, setting D ′ = pro j D (Γ ′ ) , w e ma y no w view Γ ′ as an irreduible lattie in S ′ × D ′ . W e ma y th us apply Theorem 5.18 . Notie that the same argumen t as b efore sho ws that D ′ has trivial amenable radial. W e laim that the anoni al disr ete kernel Γ ′ D ′ is in fat a dir e t fator of D ′ . Indeed, sine Γ is residually nite b y Malev's theorem [ Mal40 ℄, Prop osition 4.9 ensures that Γ ′ D ′ en tralises the disrete residual D ′ ( ∞ ) . In partiular D ′ ( ∞ ) ∩ Γ ′ D ′ = 1 sine D ′ has trivial amenable radial. F urthermore, sine D ′ / Γ ′ D ′ is a semi-simple group, its disrete residual has nite index. In partiular, up on replaing D ′ b y a nite index subgroup w e ha v e D ′ ∼ = D ′ ( ∞ ) × Γ ′ D ′ as desired. It also follo ws that Γ ′ D ′ itself admits a Zariski-dense represen tation in a semi-simple group. It remains to onsider again the pro jetion maps pro j D i : D → D i . Restriting these maps to D ′ and using the fat that pro j D i ( D ′ ) is dense, w e obtain that D i ∼ = pro j D i ( D ′ ( ∞ ) ) × D ′ i , where D ′ i = pro j D i (Γ ′ D ′ ) . The nal onlusion follo ws b y applying Corollary 5.21 to the irreduible lattie Γ ′ D ′ < D ′ 1 × D ′ 2 . Pr o of of Cor ol lary 5.3 . Sine Γ is nitely generated and irreduible, all G i are ompatly generated (alternativ ely , apply Lemma 2.12 ). W e laim that all pro jetions Γ → G i are injetiv e. Indeed, if not, then (b y indution on n ) there is j su h that the anonial disrete k ernel Γ G j is non-trivial. It is then o ompat, whi h implies that the pro jetion G j / Γ G j × Y i 6 = j G i − → Y i 6 = j G i ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 37 is prop er. This is on tradits the fat that the image of Γ in the left hand side ab o v e is disrete whilst it is dense in the righ t hand side, pro ving the laim. Supp ose giv en a linear represen tation of Γ in harateristi 6 = 2 , 3 whose image is not virtually soluble. Arguing as in [ Mon05 ℄, w e an redue to the ase where w e ha v e a Zariski- dense represen tation τ : Γ → H ( K ) in a non-trivial onneted adjoin t absolutely simple group H o v er a nitely generated eld K . Sine τ (Γ) is innite, w e an ho ose a ompletion k of K for whi h τ (Γ) is un b ounded [ BG07 , 2.1℄. P art of the argumen t in [ Mon05 ℄ is dev oted to pro ving that the represen tation is a p os- teriori faithful. One an adapt the en tire pro of to the presen t setting, but w e prop ose an alternativ e line of reasoning using an amenabilit y theorem from [ BS06 ℄. Supp ose to w ards a on tradition that the k ernel Γ 0 ✁ Γ of τ is non-trivial. Sine the pro jetions are in- jetiv e, the losure N i of the image of Γ 0 in G i is a non-trivial losed subgroup, whi h is normal b y irreduibilit y and hene is o ompat. Then Theorem 1.3 in [ BS06 ℄ implies that Γ / Γ 0 is amenable, on traditing the fat that τ (Γ) is not virtually soluble in view of Tits' alternativ e [ Tit72 ℄. A t this p oin t w e an onlude b y Theorem 5.1 . 6. Geometri arithmetiity 6.A. CA T(0) latties and parab oli isometries. W e no w sp eialise the v arious arith- metiit y results of Setion 5 to the ase of latties in CA T(0) spaes and om bine them with some of our geometri results. Reall that a parab oli isometry is alled neutral if it has zero translation length; the follo wing on tains Theorem 1.7 from the In tro dution. Theorem 6.1. L et X b e a pr op er CA T(0) sp a e with o omp at isometry gr oup and Γ < G := Is( X ) b e a nitely gener ate d latti e. Assume that Γ is irr e duible and that G ontains a neutr al p ar ab oli isometry. Then one of the fol lowing assertions holds: (i) G is a non- omp at simple Lie gr oup of r ank one with trivial entr e. (ii) Ther e is a sub gr oup Γ D ⊆ Γ normalise d by G , whih is either nite or innitely gener ate d and suh that the quotient Γ / Γ D is an arithmeti latti e in a pr o dut of semi-simple Lie and algebr ai gr oups. Pr o of. Let X ′ ⊆ X b e the anonial subspae pro vided b y Theorem 3.11 ; notie that X ′ still admits a neutral parab oli isometry . Theorem 1.6 in [ CM08 ℄ and its addendum no w apply to X ′ . The spae X ′ has no Eulidean fator: indeed, otherwise Theorem 3.8 w ould imply X ′ = R , whi h has no parab oli isometries. The k ernel of the Γ -ation on X ′ is nite and w e will inlude it in the subgroup Γ D b elo w. W e distinguish t w o ases aording as X ′ has one or more fators. In the rst ase, Is( X ′ ) annot b e totally disonneted sine otherwise Corollary 6.3 in [ CM08 ℄ p oin t ( i ) rules out neutral parab oli isometries. Th us Is( X ′ ) is a non-ompat simple Lie group with trivial en tre. If its real rank is one, w e are in ase (i); otherwise, Γ is arithmeti b y Margulis' arithmetiit y theorem [ Mar91 , Theorem 1.(1')℄ and w e are in ase (ii). F or the rest of the pro of w e treat the ase of sev eral fators for X ′ ; let Γ ∗ and let H Γ ∗ b e as in Setion 4.B . Note that H Γ ∗ ats o ompatly on ea h irreduible fator of X ′ . F urthermore, ea h irreduible fator of H Γ ∗ is non-disrete b y Theorem 4.2 . Therefore H Γ ∗ is a pro dut of the form S × D (p ossibly with one trivial fator), where S is a semi-simple Lie group with trivial en tre and D is a ompatly generated totally disonneted group without disrete fator. 38 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD By Corollary 6.3 in [ CM08 ℄ p oin t ( i ) , the existene of a neutral parab oli isometry in G implies that Is( X ′ ) is not totally disonneted. Lemma 4.8 ensures that the iden tit y omp onen t of Is( X ′ ) is in fat on tained in H Γ ∗ . Therefore, up on passing to a nite index subgroup, the iden tit y omp onen t of Is( X ′ ) oinides with S . If D is trivial, then H Γ ∗ = S is a onneted semi-simple Lie group on taining Γ as an irreduible lattie. Sine S is non-simple, it has higher rank and w e ma y app eal again to Margulis' arithmetiit y theorem; th us w e are done in this ase. Otherwise, D is non-trivial and w e ma y then apply Theorem 5.18 . It remains to he k that the normal subgroup Γ D < Γ , if non-trivial, is not nitely generated. But w e kno w that Γ D is a disrete normal subgroup of D . By Theorem 2.4 , the lattie Γ , and hene also H Γ ∗ , ats minimally without xed p oin t at innit y on ea h irreduible fator of X ′ . Therefore, Corollary 5.8 in [ CM08 ℄ ensures that D has no nitely generated disrete normal subgroup, as desired. Here is another v ariation, of a more geometri a v our; this time, it is not required that there b e a neutr al parab oli isometry: Theorem 6.2. L et X b e a pr op er ge o desi al ly omplete CA T(0) sp a e with o omp at isom- etry gr oup and Γ < Is( X ) b e a nitely gener ate d latti e. Assume that Γ is irr e duible and r esidual ly nite. If G := Is ( X ) ontains any p ar ab oli isometry, then X is a pr o dut of symmetri sp a es and BruhatTits buildings. In p artiular, Γ is an arithmeti latti e unless X is a r e al or omplex hyp erb oli sp a e. Pr o of. W e main tain the notation of the previous pro of and follo w the same argumen ts. W e do not kno w a priori whether there exists a neutral parab oli isometry . Ho w ev er, under the presen t assumption that X is geo desially omplete, Corollary 6.3 in [ CM08 ℄ p oin t ( iii ) sho ws that the existene of any parab oli isometry is enough to ensure that Is( X ′ ) is not totally disonneted. Th us the onlusion of Theorem 6.1 holds. In ase (i), Theorem 7.4 in [ CM08 ℄ p oin t ( iii ) ensures that X is a rank one symmetri spae and w e are done. W e no w assume that (ii) holds and dene D as in the pro of of Theorem 6.1 . The anonial disrete k ernel Γ D is trivial b y Theorem 4.11 . Sine D has no non-trivial ompat normal subgroup b y Corollary 5.8 in [ CM08 ℄, it follo ws from Theorem 5.18 that D is a totally disonneted semi-simple algebrai group. Therefore, the desired result is a onsequene of Theorem 7.4 in [ CM08 ℄ p oin t ( iii ). F or the reord, w e prop ose a v arian t of Theorem 6.2 : Theorem 6.3. L et X b e a pr op er ge o desi al ly omplete CA T(0) sp a e with o omp at isom- etry gr oup and Γ < Is( X ) b e a nitely gener ate d latti e. Assume that Γ is irr e duible and that every normal sub gr oup of Γ is nitely gener ate d. If G := Is ( X ) ontains any p ar ab oli isometry, then X is a pr o dut of symmetri sp a es and BruhatTits buildings of total r ank ≥ 2 . In p artiular, Γ is an arithmeti latti e. Pr o of. As for Theorem 6.2 , w e an apply Theorem 6.1 . W e laim that ase (i) is ruled out under the urren t assumptions. Indeed, a lattie in a simple Lie group of rank one is relativ ely h yp erb oli (see [ F ar98 ℄ or [ Osi06 ℄) and as su h has n umerous innitely generated normal subgroups (and is ev en SQ-univ ersal, see [ Gro87 ℄ or [ Del96 ℄ for the h yp erb oli ase and [ AMO07 ℄ for the general relativ e ase). In ase (ii) the disrete k ernel Γ D is trivial and rank one is exluded as in ase (i) if the group is Ar himedean; if it is non-Ar himedean, then there are no non-uniform nitely generated latties (see [ BL01 ℄) and th us Γ is again Gromo v-h yp erb oli whi h on tradits the assumption on normal, subgroups as b efore. ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 39 W e an no w omplete the pro of of some results stated in the In tro dution. Pr o of of The or em 1.5 . If Γ is residually nite, then Theorem 6.2 yields the desired onlu- sion; it therefore remains to onsider the ase where Γ is not residually nite. W e follo w the b eginning of the pro of of Theorem 6.2 un til the in v o ation of Theorem 4.11 , sine the latter no longer applies. Ho w ev er, w e still kno w that there is a non-trivial Lie fator in Is( X ′ ) and therefore w e apply Theorem 5.18 in order to obtain the desired onlusion ab out the lattie Γ . As for the symmetri spae fator of the spae, it is pro vided b y Theorem 7.4 in [ CM08 ℄ p oin t ( iii ) . Pr o of of Cor ol lary 1.6 . One impliation is giv en b y Theorem 1.5 . F or the on v erse, it sues to reall that unip oten t elemen ts exist in all semi-simple Lie groups of p ositiv e real rank. 6.B. Arithmetiit y of linear CA T(0) latties. W e start b y onsidering CA T(0) latties with a linear non-disrete linear ommensurator: Theorem 6.4. L et X b e a pr op er ge o desi al ly omplete CA T(0) sp a e with o omp at isom- etry gr oup and Γ < Is( X ) b e a nitely gener ate d latti e. Assume that Is( X ) p ossesses a nitely gener ate d sub gr oup Λ ontaining Γ as a sub gr oup of innite index, and ommensu- r ating Γ . If X is irr e duible and Λ p ossesses a faithful nite-dimensional line ar r epr esentation (in har ateristi 6 = 2 , 3 ), then X is a symmetri sp a e or a BruhatTits building; in p artiular Γ is an arithmeti latti e. Remark 6.5. Sev eral examples of irreduible CA T(0) spaes X of dimension > 1 ad- mitting a disrete o ompat group of isometries with a non-disrete ommensurator in Is( X ) ha v e b een onstruted b y F. Haglund [ Hag98 ℄ and A. Thomas [ Tho06 ℄ (see also [ Hag , Théorème A℄ and [ BT ℄). In all ases that spae X is endo w ed with w alls; in partiular X is the union of t w o prop er losed on v ex subspaes. This implies in partiular that X is not a Eulidean building. Therefore, Theorem 6.4 has the follo wing onsequene: in the aforemen tioned examples of Haglund and Thomas, either the ommensur ator of the latti e is nonline ar, or it is the union of a tower of latti es . In fat, as omm uniated to us b y F. Haglund, for most of these latties the ommensurator on tains ellipti elemen ts of in- nite order; this implies righ t a w a y that the ommensurator is not an asending union of latties and, hene, it is nonlinear. Note on the other hand that it is already kno wn that Is( X ) is mostly nonlinear in these examples, sine it on tains losed subgroups isomorphi to the full automorphism group of regular trees. Pr o of of The or em 6.4 . Sine X is irreduible and the ase X = R satises the onlusions of the theorem, w e assume heneforth that X has no Eulidean fator. The Is( X ) -ation on X is minimal b y Lemma 3.13 in [ CM08 ℄ and has no xed p oin t at innit y b y Corollary 3.12 . In partiular, w e an apply Theorem 1.1 in [ CM08 ℄: either Is( X ) is totally disonneted or it is simple Lie group with trivial en tre and X is the asso iated symmetri spae. In the latter ase, Margulis' arithmetiit y theorem nishes the pro of. W e assume heneforth that Is( X ) is totally disonneted. Let G denote the losure of Λ in Is( X ) . Note that G ats minimally without xed p oin t at innit y , sine it on tains a subgroup, namely Γ , whi h p ossesses these prop erties b y Theorem 2.4 . In partiular G has trivial amenable radial b y Theorem 1.10 in [ CM08 ℄ and th us the same holds for the dense subgroup Λ < G . In partiular an y faithful represen tation of Λ to an algebrai group yields a faithful represen tation of Λ to an adjoin t semi-simple algebrai group with Zariski-dense image, to whi h w e an apply Theorem 5.17 . As w e sa w, 40 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD the group G has no non-trivial ompat (in fat amenable) normal subgroup and furthermore G is irreduible sine X is so, see Theorem 1.10 in [ CM08 ℄. The fat that the lattie Γ has innite index in Λ rules out the disrete ase. Therefore G is a simple algebrai group and Γ an arithmeti lattie. It remains to dedue that X has the desired geometri shap e. This will follo w from Theorem 7.4 in [ CM08 ℄p oin t ( iii ) pro vided w e sho w that ∂ X is nite-dimensional and that G has full limit set. The rst fat holds sine X is o ompat; the seond is pro vided b y Corollary 2.10 . Remark 6.5 illustrates that Theorem 6.4 fails dramatially if one assumes only that Γ is linear. Ho w ev er, passing no w to the ase where X is reduible, the linearit y of Γ is enough to establish arithmetiit y , indep enden tly of an y assumption on ommensurators, the result announed in Theorem 1.8 in the In tro dution. Theorem 6.6. L et X b e a pr op er ge o desi al ly omplete CA T(0) sp a e with o omp at isom- etry gr oup and Γ < Is( X ) b e a nitely gener ate d latti e. Assume that Γ is irr e duible and p ossesses some faithful line ar r epr esentation (in har ateristi 6 = 2 , 3 ). If X is r e duible, then Γ is an arithmeti latti e and X is a pr o dut of symmetri sp a es and BruhatTits buildings. Pr o of. In view of Theorem 3.8 , w e an assume that X has no Eulidean fator. The Is( X ) - ation on X is minimal b y Lemma 3.13 in [ CM08 ℄ and has no xed p oin t at innit y b y Corollary 3.12 . In partiular, w e an apply Theorem 1.1 in [ CM08 ℄ to obtain deomp osi- tions of Is( X ) and X in whi h the fators of X orresp onding to onneted fators of Is( X ) are isometri to symmetri spaes. There is no loss of generalit y in assuming Γ ∗ = Γ in the notation of Setion 4.B . Let no w G b e the h ull of Γ . By Remark 4.6 , the group Γ is an irreduible lattie in G . Sine Is( X ) ats minimally without xed p oin t at innit y , it follo ws from Corollary 2.7 that Γ has trivial amenable radial. In partiular an y faithful represen tation of Γ to an algebrai group yields a faithful represen tation of Γ to an adjoin t semi-simple algebrai group with Zariski-dense image, to whi h w e an apply Theorem 5.1 . The group G has no non-trivial ompat normal subgroup e.g. b y minimalit y . F urther- more the disrete fator is trivial b y Theorem 4.2 . Therefore G is a simple algebrai group and Γ an arithmeti lattie. It remains to dedue that X has the desired geometri shap e and this follo ws exatly as in the pro of of Theorem 6.4 . 6.C. A family of examples. W e shall no w onstrut a family of latties Γ < G = S × D as in the statemen t of Theorem 5.18 (see also Theorem 6.2 ) with the follo wing additional prop erties: (i) There is a prop er CA T(0) spae Y with D < Is( Y ) su h that the D -ation is o ompat, minimal and without xed p oin t at innit y . In partiular, setting X = X S × Y , where X S denotes the symmetri spae asso iated to S , the Γ -ation on X is prop erly dison tin uous (in fat free), o ompat, minimal, without xed p oin t at innit y . (ii) The anonial disrete k ernel Γ D ✁ D is innite (in fat, it is a free group of oun table rank). (iii) The pronite k ernel of D / Γ D → Q is non-trivial. Remark 6.7. Sine D is minimal, it has no ompat normal subgroup and th us w e see that the pronite extension app earing in Theorem 5.18 annot b e eliminated. ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 41 W e b egin with a general onstrution: Let g b e (the geometri realisation of ) a lo ally nite graph (not redued to a single p oin t) and let Q < Is( g ) a losed subgroup whose ation is v ertex-transitiv e. In partiular, Q is a ompatly generated totally disonneted lo ally ompat group. W e p oin t out that an y ompatly generated totally disonneted lo ally ompat group an b e realised as ating on su h a graph b y onsidering S hreier graphs g , see [ Mon01 , 11.3℄; the k ernel of this ation is ompat and arbitrary small. On the other hand, if Q is a non-Ar himedean semi-simple group, one an also tak e v ery expliit graphs dra wn on the BruhatTits building of Q , e.g. the 1 -sk eleton (this part is inspired b y [ BM00a , 1.8℄, see also [ BMZ04 ℄). Let moreo v er C b e an innite pronite group and ho ose a lo ally nite ro oted tree t with a lev el-transitiv e C -ation for whi h ev ery innite ra y has trivial stabiliser. F or instane, one an ho ose the oset tree asso iated to an y nested sequene of op en subgroups with trivial in tersetion, see the pro of of Théorème 15 in 6 on p. 82 in [ Ser77 ℄. W e dene a lo ally nite graph h with a C × Q -ation as the 1 -sk eleton of the square omplex t × g . Let a = e h b e the univ ersal o v er of h , Λ = π 1 ( h ) and dene the totally disonneted lo ally ompat group D b y the orresp onding extension 1 − → Λ − → D − → C × Q − → 1 . Prop osition 6.8. Ther e exists a pr op er CA T(0) sp a e Y suh that D sits in Is( Y ) as a lose d sub gr oup whose ation is o omp at, minimal and without xe d p oint at innity. Pr o of. One v eries readily the follo wing: Lemma 6.9. L et a b e (the ge ometri r e alisation of ) a lo al ly nite simpliial tr e e and D < Is( a ) any sub gr oup. L et x ∈ a b e a vertex and let Y b e the ompletion of the metri sp a e obtaine d by assigning to e ah e dge of a the length 2 − r , wher e r is the ombinatorial distan e fr om this e dge to the ne ar est p oint of the orbit D .x . Then Y is a pr op er CA T(0) sp a e with a o omp at ontinuous isometri D -ation. Mor e- over, if the D -ation on a was minimal or without xe d p oint at innity, then the orr e- sp onding statement holds for the D -ation on Y . Apply the lemma to the tree a = e h onsidered earlier. W e laim that the D -ation on a is minimal. Clearly it sues to sho w that the Λ -ation is minimal. Note that Λ ats transitiv ely on ea h bre of p : e h → h . Th us it is enough to sho w that the on v ex h ull of a giv en bre meets ev ery other bre. Consider t w o distint v erties v , v ′ ∈ h . The pro dut nature of h mak es it lear that v and v ′ are b oth on tained in a ommon minimal lo op based at v . This lo op lifts to a geo desi line in e h whi h meets the resp etiv e bres of v and v ′ alternativ ely and p erio dially . In partiular, this onstrution yields a geo desi segmen t joining t w o p oin ts in the bre ab o v e v and on taining a p oin t sitting ab o v e v ′ , whene the laim. Sine Λ ats freely and minimally on the tree a whi h is not redued to a line, it follo ws that Λ xes no end of a . Th us the lemma pro vides a prop er CA T(0) spae Y with a o ompat minimal isometri D -ation, without xed p oin t at innit y . It remains to sho w that D < Is( Y ) is losed. This holds b eause the totally disonneted groups Is( a ) and Is( Y ) are isomorphi; indeed, the anonial map a → Y indues a on tin uous surjetiv e homomorphism Is( a ) → Is( Y ) , whi h is th us op en. Remark 6.10. The ab o v e onstrution giv es an example of a prop er CA T(0) spae with a totally disonneted o ompat and minimal group of isometries su h that not all p oin t stabilisers are op en. Consider indeed the p oin ts added when ompleting. Their stabilisers 42 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD map to Q under D → ( C × Q ) and hene annot b e op en. In other w ords, the ation is not smo oth in the terminology of [ Cap07 ℄. Notie ho w ev er that the set of p oin ts with op en stabiliser is neessarily a dense on v ex in v arian t set. W e shall no w sp eialise this general onstrution to yield our family of examples. Let K , H , Σ , K (Σ) b e as desrib ed after Theorem 5.1 on p. 25 . W e write Σ f , Σ ∞ ⊆ Σ for the subsets of nite/innite plaes and assume that b oth are non-empt y . Let S = Q v ∈ Σ f H ( K v ) + and Q = Q v ∈ Σ ∞ H ( K v ) + . The group ∆ = H ( K (Σ)) ∩ ( S × Q ) is an irreduible o ompat lattie in S × Q . Let C b e an y pronite group with a dense inlusion ∆ → C . W e no w em b ed ∆ diagonally in S × C × Q ; learly ∆ is a o ompat lattie. Let Γ < G = S × D b e its pre-image. Then Γ is a o ompat lattie sine it on tains the disrete k ernel of the anonial map G → S × C × Q . It is learly irreduible and therefore pro vides an example that the struture of the desription in the onlusion of Theorem 5.18 annot b e simplied. F urthermore, the normal subgroup app earing in Theorem 6.1 (ii) is also una v oidable. W e end this setion with a few supplemen tary remarks on the preeding onstrution: (i) If the pronite group C has no disrete normal subgroup, then Γ D = π 1 ( h ) oinides with the quasi-en tre of D . This w ould b e the ase for example if C = H ( K v ) and H is almost K -simple of higher rank, where v is a non-Ar himedean v aluation su h that H is K v -anisotropi. In partiular, in that situation Γ D is the unique maximal disrete normal subgroup of D and the quotien t D / Γ D has a unique maximal ompat normal subgroup. Th us the group G admits a unique deomp osition as in Theorem 5.18 in this ase. (ii) W e emphasise that, ev en though D / Γ D deomp oses as a diret pro dut C × Q in the ab o v e onstrution, the group D admits no non-trivial diret pro dut de- omp osition, sine it ats minimally without xed p oin t at innit y on a tree (see Theorem 1.10 in [ CM08 ℄). (iii) The fat that D / Γ D deomp oses as a diret pro dut C × Q is not a oinidene. In fat, this is happ ens alw a ys pro vided that ev ery o ompat lattie in S has the Congruene Subgroup Prop ert y (CSP). Indeed, giv en a ompat op en subgroup U of D / Γ D , the in tersetion Γ U of Γ ∩ ( S × U ) is an irreduible lattie in S × U with trivial anonial disrete k ernels. By (CSP), up on replaing Γ U b y a nite index subgroup (whi h amoun ts to replae U b y an op en subgroup), the pronite ompletion c Γ U splits as the pro dut o v er all primes p of the pro- p ompletions [ (Γ U ) p , whi h are just-innite. Th us the anonial surjetiv e map c Γ U → U sho ws that U is a diret pro dut. This implies that the maximal ompat normal subgroup of D / Γ D is a diret fator. (iv) A ording to a onjeture of Serre's (fo otnote on page 489 in [ Ser70 ℄), if S has higher rank then ev ery irreduible lattie in S has (CSP). (See [ Rag04 ℄ for a reen t surv ey on this onjeture.) 7. A few questions W e onlude b y olleting some further questions that w e ha v e enoun tered while w orking on this pap er. It is w ell kno wn that the Tits b oundary of a prop er CA T(0) spae with o ompat isometry group is neessarily nite-dimensional (see [ Kle99 , Theorem C℄). It is quite p ossible that the same onlusion holds under a m u h w eak er assumption. ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: DISCRETE SUBGR OUPS 43 Question 7.1. Let X b e a prop er CA T(0) spae su h that Is( X ) has full limit set. Is the b oundary ∂ X nite-dimensional? A p ositiv e answ er to this question w ould sho w in partiular that the seond set of as- sumptions denoted (b) in Theorem 7.4 in [ CM08 ℄ is in fat redundan t. Let G b e a simple Lie group ating on tin uously b y isometries on a prop er CA T(0) spae X . The Karp elevi hMosto w theorem ensures that there exists a on v ex orbit when X is a symmetri spae of non-ompat t yp e. This statemen t, ho w ev er, annot b e generalised to arbitrary X in view of Example 7.7 in [ CM08 ℄. Question 7.2. Let G b e a simple Lie group ating on tin uously b y isometries on a prop er CA T(0) spae X . If the ation is o ompat, do es there exist a on v ex orbit? It is sho wn in Theorem 7.4 in [ CM08 ℄ p oin t ( iii ) that if X is geo desially omplete, then the answ er is p ositiv e. It is go o d to k eep in mind Example 7.6 in [ CM08 ℄, whi h sho ws that the natural analogue of this question for a simple algebrai group o v er a non- Ar himedean lo al eld has a negativ e answ er. More optimistially , one an ask for a on v ex orbit whenev er the simple Lie group ats on a omplete (not neessarily prop er) CA T(0) spae, but assuming the ation non-ev anesen t (in the sense of [ Mon06 ℄). A p ositiv e answ er w ould imply sup errigidit y statemen ts up on applying it to spaes of equiv arian t maps. Man y of our statemen ts on CA T(0) latties require the assumption of nite generation. One should of ourse w onder for ea h of them whether it remains v alid without this as- sumption. One instane where this question is esp eially striking is the follo wing (see Corollary 3.12 ). Question 7.3. Let X b e a prop er CA T(0) spae whi h is minimal and o ompat. Assume that Is( X ) on tains a lattie. Is it true that Is( X ) has no xed p oin t at innit y? In a forthoming artile [ CM08b ℄, w e shall establish a p ositiv e answ er to this question b y in v estigating the rle of unimo dularit y for the full isometry group. W e ha v e seen in Corollary 7.12 in [ CM08 ℄ that if the isometry group of a prop er CA T(0) spae X is non-disrete in a strong sense, then Is( X ) omes lose to b eing a diret pro dut of top ologially simple groups. Question 7.4. Retain the assumptions of Corollary 7.12 in [ CM08 ℄. Is it true that so c( G ∗ ) is a pro dut of simple groups? Is it o ompat in G , or at least do es G ha v e ompat Ab elianisation? Clearly Corollary 7.12 in [ CM08 ℄ redues the question to the ase where Is( X ) is totally disonneted. One an also ask if so c( G ∗ ) is ompatly generated (whi h is the ase e.g if it is o ompat in G ). If so, w e obtain additional information b y applying Prop osition 6.12 in [ CM08 ℄. In the ab o v e situation one furthermore exp ets that the geometry of X is eno ded in the struture of Is( X ) . In preise terms, w e prop ose the follo wing. Question 7.5. Retain the assumptions of Corollary 7.12 in [ CM08 ℄. It it true that an y prop er o ompat ation of G on a prop er CA T(0) spae Y yields an equiv arian t isometry ∂ X → ∂ Y b et w een the Tits b oundaries? Or an equiv arian t homeomorphism b et w een the b oundaries with resp et to the ne top ology? The disussion around Corollary 5.3 ( f. Remark 5.4 ) suggests the follo wing. 44 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Question 7.6. Let Γ < G = G 1 × · · · × G n b e an irreduible nitely generated lattie, where ea h G i is a lo ally ompat group. Do es ev ery harater Γ → R extend on tin uously to G ? Y. Shalom [ Sha00 ℄ pro v ed that this is the ase when Γ is o ompat and in some other situations. Referenes [AB94℄ Norb ert A'Camp o and Mar Burger, R ése aux arithmétiques et ommensur ateur d'apr ès G. A. Mar gulis , In v en t. Math. 116 (1994), no. 1-3, 125. [AB98℄ Sot A dams and W erner Ballmann, A menable isometry gr oups of Hadamar d sp a es , Math. Ann. 312 (1998), no. 1, 183195. 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