A spectral sequence to compute L2-Betti numbers of groups and groupoids

A SPECTRAL SEQUENCE TO COMPUTE L 2 -BETTI NUMBERS OF GR OUPS AND GR OUPOIDS ROMAN SAUER AND ANDREAS THOM Abstract. W e construct a spectral sequence for L 2 -ty p e cohomology groups of discr ete measured groupoids. Based on the sp ectral sequence, we prov e the Hopf-Singer conject ure for aspherical manifolds with p oly-surface fundamental groups. More generally , w e obtain a permanence r esult for the Hopf-Singer conjecture under taking ﬁber bundles whose base space is an aspherical manifold with p oly-surf ace fundamen ta l group. As further sample applications of the sp ectral sequence, we obtain new v anishing theorems and explicit computations of L 2 -Betti num bers of groups and manifolds and obstructions to the existence of norm al subrelations in measured equiv alence relations. 1. I ntroduction and S t a tement of Resul ts The aim of this work is to construct, starting with a shor t exact sequence (later called str ong extension ) of discre te measured gr oup oids, a sp ectral sequence for L 2 -type c ohomol- ogy gr o ups. F or this, we ar e us ing a blend of to ols from homolo gical algebra a nd ergo dic theory . Gab oriau introduced and studied the notion o f L 2 -Betti num bers of measured e q uiv a le nc e relations [ 13 ], which prov ed to b e very fruitful, e sp e c ially in a pplications to von Neumann algebras via the work of Popa. Subsequently , mo r e algebra ic deﬁnitions w ere developed [ 23 , 28 ] that build up on L¨ uck’s algebra ic theor y o f L 2 -Betti num bers [ 2 1 ]. W e work with the latter since they are e s pec ially suited for our pur po ses. Gab oriau’s and L¨ uck’s works add quite diﬀeren t computational tec hniques to the theory of L 2 -Betti num ber s: F or instance, Gab o r iau’s theory a llows to exploit the fa c t that the erg o dic dimension of a gr oup might b e muc h smaller than its coho mo logical dimension; L¨ uc k’s algebraic theory , on the other hand, allows to use the power of standa rd homologic al a lgebra in computations of L 2 -Betti n umbers (another algebraic L 2 -theory is due to F arb er [ 8 ]). The motiv ation for our s p ectr al sequence w as to combine these computational adv an tages. The rea de r may w onder whether the gener ality of the la nguage of group oids is necessary; we will pres ent computations (Corollar ies 1.8 , 1.13 , 1.15 ) for groups and manifolds that, in their pro ofs, make use of measur e d equiv alence r elations. The cla ss of measured equiv a lence relations, howev er, is not clos ed under tak ing quo tien ts, unlike the cla ss of discrete measured group oids; so it turns out that it is necess ary a nd most natural to work with group oids in our con text. 2000 Mathematics Subje c t Classiﬁc ation. Primary 37A20, 46L99; Secondary 18B40, 18G40. R.S. ackno wledges supp ort by DFG gran t SA 1661/1-1 and thanks the Max-Pl anck-Institute in Bonn for its hospitality during the ini tial stage of this pro ject. A.T. thanks the Graduiertenko lleg ”Grupp en und Geometrie”, G¨ ottingen. 1 2 R OMAN S AUER AND ANDREAS THOM W e r efer the rea der for a more detailed review o f used to ols, concepts, a nd notation to Section 2 . A cen tral notion is that o f a str ongly normal su b gr oup oid of a discr ete measured group oid. Building on [ 9 ], we see in Section 3 that a stro ngly normal subgroup oid (Deﬁnition 3.5 ) a llows a quotient construction, and every strongly normal subg roup oid app ears a s the kernel of a quotient ma p. One calls an erg o dic G a st ro ng extension of S and Q if S ⊂ G is a strongly normal subgroup oid and Q the co rresp onding quotient gr oup oid. Several exa mples of strong extensions are discussed in Sectio n 5 . F or a discrete mea sured gr oup oid G , we deﬁne cohomology g roups H ∗ ( G , M ) with co ef- ﬁcient s in a mo dule M over the gro up o id ring of G (s e e Section 4 ). If M is the algebr a of aﬃliated op erator s U ( G ) o f the v on Neumann a lgebra o f G , then L ¨ uc k’s dimension o f the U ( G )-mo dule H n ( G , U ( G )) deﬁnes the n -th L 2 -Betti num ber b (2) n ( G ) of G . This deﬁnition of b (2) n ( G ) co incides with the deﬁnition given by the ﬁrst author in [ 23 ] which itself is an algebraic formulation of Gab or ia u’s deﬁnition [ 13 ]. Although it gives the same L 2 -Betti nu mbers, our deﬁnition of H ∗ ( G , M ) is mor e inv olved: we hav e to take, in so me se ns e, a hu ge mode l o f H ∗ ( G , M ) to o vercome some serious, tec hnical diﬃculties. Theorem 4.7 of Section 4 is our main re s ult: a Grothendieck sp ectral sequence that computes the cohomolog y of a str o ng extension of S and Q in terms of the cohomolo gy groups of S a nd Q . W e now pr esent applicatio ns o f the sp ectr al s equence. The pro o fs of the theorems below are found in Sectio n 7 . 1.1. Applications to ergo dic theory. The following theorem generalizes a co r resp onding result by Gab or iau [ 13 , Th ´ eor` eme 6.6 ] for e xtension of groups where the quotient group is amenable. Theorem 1.1. L et 1 → S − → G − → Q → 1 b e a str ong extension of discr ete me asur e d gr oup oids wher e Q is an inﬁnite amenable discr ete me asur e d gr oup oid. If b (2) n ( S ) < ∞ , then b (2) n ( G ) = 0 . Corollary 1.2. L et Γ b e a c ountable gr oup with b (2) 1 (Γ) 6 = 0 . Then ther e is a fr e e, er go dic Γ - pr o b abi lity sp a c e such that the asso ciate d orbit e quivalenc e r ela tion R has a str ongly normal subr elation S ⊂ R of inﬁ nite index with b (2) 1 ( S ) = ∞ . Pr o of of Cor ol lary. Because of b (2) 1 (Γ) 6 = 0 the gr oup Γ do e s not hav e prop erty ( T ). By [ 24 , Theorem 1.5] there is an ergo dic Γ-probability spa ce ( X, µ ) that is not str ongly er go dic (deﬁned in [ 24 , Section 1]). By [ 24 , Theor ems 2.1 and 2 .3] (the details o f the proof are in [ 18 ]), there is an amena ble equiv alence relation R hyp and a strong sur jection θ : X ⋊ Γ → R hyp . The claim no w follo ws for S = ker( θ ) from Theor em 1.1 and b (2) p ( X ⋊ Γ) = b (2) p (Γ) (see Section 4.5 ).  Theorem 1 . 3. If an er go d ic discr ete me asur e d gr oup oi d G p ossesses a str ongly normal sub gr oup oid S such that b (2) n ( S ) = 0 for every n ≤ d , then b (2) n ( G ) = 0 for every n ≤ d . The question of (non-)existence o f subrelations that are (strong ly) normal or split oﬀ as a factor is inv estigated in sev eral works. F or example, in [ 9 , Theorem 4.4] it is proved that orbit equiv a le nce re la tions of free I I 1 -actions o f a lattice in a simple, connected, non-compact A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 3 Lie group never have inﬁnite, strongly normal, amenable subrelations . In [ 1 , Corolla ry 6.2] it is shown that the orbit equiv alence relation of a type II 1 -action of a non-e lement ary word-h ype rb olic g roup is irre ducible, that is, it never splits o ﬀ an inﬁnite, discrete measured equiv a lence relation as a facto r. Corollary 1.4. L et G b e an inﬁnite, amenable, er go d ic discr ete me asur e d gr oup oid. T hen b (2) p ( G ) = 0 for every p ≥ 0 . Pr o of of Cor ol lary. Consider the principal ex tension 1 → G stab → G → G rel → 1 o f Sub- section 5 .3 . Assume ﬁrst that G stab is inﬁnite. Since G is amenable, the stabilizer gr oups G x = { γ ∈ G ; r ( γ ) = s ( γ ) = x } are (inﬁnite) amenable for a.e . x ∈ G 0 . By Lemma 5.1 , we hav e b (2) p ( G stab ) = 0 for every p ≥ 0 . B y Theore m 1.3 the assertion follo ws. Suppo se now that G stab is ﬁnite. Then G rel has to b e inﬁnite since G is so. Hence G rel is an inﬁnite h ype r ﬁnite equiv alence relation. Now the assertion is implied by Lemma 5.1 and Theorem 1.1 .  Theorem 1.5. L et 1 → S − → G − → Q → 1 b e a stro ng extension of discr ete m e a sur e d gr oup o ids wher e Q is an inﬁnite me asur e d gr oup oid. Assume that b (2) p ( S ) = 0 for al l 0 ≤ p ≤ d and b (2) d +1 ( S ) < ∞ . Then b (2) p ( G ) = 0 for al l 0 ≤ p ≤ d + 1 . As a consequence, w e obtain an alter native pro of of [ 3 , Th ´ eor` eme 5.4]: Corollary 1.6 (Berg eron-Gab or iau) . L et Γ b e a c ountable gr o up with b (2) 1 (Γ) 6 = 0 . L et ( X , µ ) b e an er go dic Γ -pr ob ability s p ac e. Then either the stabilizer Γ x is ﬁnite fo r µ -a.e. x ∈ X , or b (2) 1 (Γ x ) = ∞ , thus Γ x is not ﬁnitely gener ate d, for µ -a.e. x ∈ X . Pr o of of Cor ol lary. By er go dicity , either Γ x is ﬁnite almost ev erywhere o r inﬁnite almost everywhere. The function x 7→ b (2) 1 (Γ x ) is measur able [ 3 , Th´ eor` eme 5.7]. Thus, by erg o dicity again, either b (2) 1 (Γ x ) = ∞ almo st everywhere or b (2) 1 (Γ x ) < ∞ almo st everywhere. Apply Theorem 1.5 to the principal extension (Subsec tion 5.3 ) in combination with Lemma 5 .1 . Note that the ﬁr st L 2 -Betti n umber of a ﬁnitely g enerated group is ﬁnite.  R emark 1.7 (Ergo dicity hypothesis) . W e remark that the notion of str ong exten s ion is based upo n er go dicity . Nevertheless, it is possible to drop the a s sumption o n er go dicity in several of the res ults abov e. In fact, if R X µ x dµ ( x ) = µ is the erg o dic decomp ositio n of ( G , µ ), then it is possible to show that b (2) p ( G , µ ) = Z X b (2) p ( G x , µ x ) dµ ( x ) . 1.2. Appli cations to L 2 -Betti num b e rs of groups and manifolds. L ¨ uc k [ 21 , The- orem 7.2 on p. 294] pr ov ed the following co rollar y under the additional assumption that the quotient group has a non- torsion elemen t or ﬁnite subgroups o f arbitrarily high o rder. Then, using equiv alence relations, Gab oriau [ 13 , Th ´ eor` eme 6.8 ] g av e a pr o of in deg ree 1, i.e. for the ﬁrst L 2 -Betti num ber , only ass uming the quotient to be inﬁnite. W e also mention that [ 6 , Cor ollary 1] gives a very elementary pr o of for the degre e 1 case if the quotient has a non-tor sion element. It is remarked therein that it is a chal lenging, and vaguely irritating question to ﬁnd a pu re ly c ohomolo gic al pr o of of Gab ori au’s re sult . Up to now, there is no 4 R OMAN S AUER AND ANDREAS THOM pro of that do es not use mea s ured equiv alence rela tions. The following coro llary of Theo- rem 1.5 g eneralizes the aforementioned results to all deg r ees without further a ssumptions on the quotien t. Corollary 1 .8. L et Λ ⊂ Γ b e a normal sub gr oup of inﬁnite inde x. Supp ose that b (2) p (Λ) = 0 for 0 ≤ p ≤ d − 1 and b (2) d (Λ) < ∞ . Then b (2) p (Γ) = 0 for 0 ≤ p ≤ d . Pr o of of Cor ol lary. One can ﬁnd free, ergo dic mea s ure-preser ving actions of Λ , Γ and Q = Γ / Λ o n pr obability spaces such that the asso c iated or bit equiv alence relations form a strong extension (see Subsection 5.2 ). Then apply Theorem 1.5 and ( 4.6 ) in Section 4 .  The no tion of me asur e e quivalenc e w as introduce d by Gr omov and, fo r the ﬁr s t time, gained prominence in the w ork o f F urman [ 10 , Deﬁnition 1.1]: Deﬁnition 1.9. Tw o co untable groups Γ and Λ a re ca lled me asur e e quivalent if there exists a non-trivia l measure space (Ω , µ ) on which Γ × Λ a cts such that the restric ted actions of Γ = Γ × { 1 } and Λ = { 1 } × Λ hav e measur able fundamen tal domains X ⊂ Ω and Y ⊂ Ω with µ ( X ) < ∞ and µ ( Y ) < ∞ . The space (Ω , µ ) is c a lled a me asu re c oupling b etw een Γ and Λ. W e denote by cd C (Γ) the c ohomo lo gic al dimension o f a group Γ ov er C , i.e. the pro jective dimension of C as a C Γ-mo dule. The following theorem is a conseq uence o f the more gener a l Theorem 7.3 and Lemma 7.2 . Theorem 1. 1 0. L et 1 → Λ → Γ → Q 0 → 1 b e a short exact se quenc e of gr oups. Supp ose that b (2) p (Λ) = 0 for p > m . L et Q 1 b e a gr o up that is me asur e e quivalent to Q 0 . L et n = cd C ( Q 1 ) . Then b (2) p (Γ) = 0 for p > n + m . R emark 1.11 . In o r der to ca pture the strength of the metho d of pr o of that leads to the ab ov e theorem, we introduce the concept of me asur abl e c ohomolo gic al dimension in Section 6 . It is then clear that the only relev ant assumption is that the measurable co homologica l dimension of the quotien t group is b ounded by n . W e will prov e this fact in Theorem 7.3 . R emark 1.12 . A typical situation wher e the quotient g roup is measure equiv a lent to a gr oup with a low er cohomologica l dimensio n is the following: Le t Γ be a co compa ct lattice in SL( n, R ). Then SL( n, R ) endow ed with the left m ultiplication action o f Γ and the rig ht m ultiplication action of SL( n, Z ) is a measur e c oupling with resp ect to the Haa r measure. The rational c ohomolog ic al dimension of Γ equa ls the dimensio n of the asso ciated symmetric space SL( n, R ) / SO( n, R ); but the rational cohomolog ical dimension of SL( n, Z ) is cd Q  SL( n, Z )  = dim  SL( n, R ) / SO( n, R )  − ( n − 1) by a result of Borel-Serre [ 5 ]. W e present the following sample application of Theo rem 1.10 , for which we do no t kno w an alternative pro o f that do es not use measured group o ids. Corollary 1.1 3. L et A 1 , . . . , A k and B 1 , . . . , B l b e inﬁnite amenable gr oups. L et Γ b e an extension of the t yp e 1 → A 1 ∗ · · · ∗ A k → Γ → B 1 ∗ · · · ∗ B l → 1 . Then b (2) p (Γ) = ( ( k − 1)( l − 1) if p = 2 , 0 otherwise. A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 5 Pr o of of Cor ol lary. Since b (2) p ( A i ) = 0 for every p ≥ 0 and i ∈ { 1 , . . . , k } by [ 21 , The- orem 6.37 on p. 2 5 9], we obtain by the Ma yer-Vietoris sequence for l 2 -homology that b (2) p ( A 1 ∗ · · · ∗ A k ) = 0 whenever p > 1. Since B 1 ∗ · · · ∗ B l and the free g roup F l of r ank l are mea sure eq uiv alent [ 14 , Section 2.2], it follows that b (2) p (Γ) = 0 for p > 2 by the pre v ious theorem. Since Γ and A 1 ∗ · · · ∗ A k are inﬁnite, w e ha ve b (2) 0 (Γ) = b (2) 0 ( A 1 ∗ · · · ∗ A k ) = 0, th us, by Coro llary 1.8 , b (2) 1 (Γ) = 0 . Since the Euler characteristic χ is multiplicative for extensions and χ (Γ) = P p ≥ 0 ( − 1) p b (2) p (Γ), we obtain that b (2) 2 (Γ) = χ (Γ) = χ ( A 1 ∗ · · · ∗ A k ) χ ( B 1 ∗ · · · ∗ B l ) = ( k − 1)( l − 1) .  Next we tur n our attention to a central conjecture for L 2 -Betti n umbers, the Hopf-Singer Conje ctu r e ; it pr edicts that for a clos ed aspherical manifold M we hav e b (2) p ( f M ) = 0 provided 2 p 6 = dim( M ). F or a survey of known res ults w e r efer the reader to [ 21 , Chapter 11]. A g r oup is sa id to b e a p oly-surfac e gr oup , if it has a se r ies of normal subgroups s uch that the subquotients are fundamen tal groups of closed orien ted sur faces (see Deﬁnition 6.10 fo r more details ). The cohomologica l dimension of such a gro up is precisely twice the length of the series and the E uler c haracteristic is the pro duct of the individual Euler c haracter istics of the subquotients. In Section 6 , w e study the measura ble co homologica l dimension (whic h is closely r elated to Ga bo riau’s e r go dic dimensio n [ 13 ]) in more detail and show, that the measurable co ho- mological dimensio n o f a po ly-surface group is the length of its deﬁning series of normal subgroups, i.e. only half of the exp ected num ber. The following theorem is obtained as Corollary 6.11 in Subsection 6.2 . Theorem 1.14. The Hopf-Singer c onje ctur e holds for any close d aspheric a l manifold with p oly-su rfac e fundamental gr oup. W e also obtain a per ma nence result for the Ho pf-Singer conjecture: Theorem 1. 1 5. L et M b e a close d, aspheric a l, 2 n -dimensional manifold that satisﬁes the Hopf-Singer Conje ctu r e, that is, b (2) p ( f M ) = 0 unless p = n . L et L b e a close d orientable aspheric al manifold of dimension 2 m with a p oly-surfac e fundamental gr oup. If N is the total sp ac e of an orientable ﬁb er bund le over L with ﬁb er M , then b (2) p ( e N ) = ( b (2) n ( f M ) · χ ( π 1 ( L )) if p = m + n , 0 otherwise. In p articular, N satisﬁes the Hopf-Singer Conje ctur e, to o. Pr o of. The ma nifo ld N is closed, orientable and as pher ical. Let Γ = π 1 ( N ), and Λ = π 1 ( M ). F r om the ﬁber bundle we get an extension of groups 1 → Λ → Γ → π 1 ( L ) → 1 . Now, π 1 ( L ) s a tisﬁes Singer ’s condition (see Deﬁnition 6.6 ) by Theor em 6.11 . Hence, the group π 1 ( L ) has measurable co ho mologica l dimensio n m . By Theorem 7.3 , b (2) p ( e N ) = b (2) p (Γ) = 0 for p > m + n . Thus, by Poincar ´ e duality , b (2) p ( e N ) = 0 unless p = m + n . This yields b (2) m + n ( e N ) = ± χ ( N ), and fro m the multip licativity o f χ for ﬁb er bundles the claim follo ws.  6 R OMAN S AUER AND ANDREAS THOM 2. Over view o f used concepts and tool s 2.1. Standard Borel and measure spaces. All measur able spaces in this w ork are st an- dar d Bor el sp ac es unless stated other w is e. Ma ps b etw een standard Borel spa ces a r e measur- able unless stated otherwise. Our background references for standard Bo r el spaces are [ 7 , 19 ]; we recall here some basic notions and facts (see also [ 15 , p. 51/52]). W e use the terms me asur able and Bor el in terchangeably fo r maps b etw een or subsets of standard Bor el s paces. A measure on a sta ndard Borel spa ce is called Bor el me asur e . A p art ition of standard Bor el s pa ce X is a countable family ( X i ) i ∈ I of pairwise disjo in t Borel subsets such that X = S i ∈ N X i . A Bor el isomorphism f : X → Y betw een standard Bor el spaces is a bijective Bo rel map. In verses of Bor el iso mo rphisms are Borel, and B orel s ubsets of a standard B o rel space are a gain standard Borel. The following result is a fundamental to ol to which we r efer throug hout the paper a s the sele ct ion the or em (see [ 19 , theorem 18.10 on p. 12 3]). Theorem 2.1 (Selec tio n Theorem) . L et f : X → Y b e a Bor el map b etwe en standar d Bor el sp ac es whose ﬁ b ers ar e c ountable. Then f ( X ) ⊂ Y is Bor el, and ther e is a p artition X = S i ∈ N X i such t hat e ach f | X i is inje ctive. A me asur e sp ac e ( X , µ ) is b y deﬁnition a standard Bo rel space X equipped with a Bo rel measure µ . A pr ob abili ty sp ac e is a measure space who se measure is a proba bilit y measure, that is, has total meas ure 1. A me asur e isomorphism f : ( X, µ X ) → ( Y , µ Y ) is a mea sure- preserving Borel map with the pro per ty that there a re Borel subsets A ⊂ X a nd B ⊂ Y with µ X ( X − A ) = µ Y ( Y − B ) = 0 such that f | A is a Borel isomorphism A → B . If ( X, µ ) is contin uous, that is , µ ( { x } ) = 0 fo r every x ∈ X , then there is Bo rel isomorphism f : X → [0 , 1] with f ∗ µ = µ ◦ f − 1 = λ . Her e, λ deno tes the Lebesgue measure. Another importa n t tool [ 15 , Theorem 3.22 on p. 72] is Theorem 2.2 (Mea sure disintegration) . L et ( X , µ ) and ( Y , ν ) b e pr ob ability s p ac es and π : X → Y a Bor el map such that π ∗ µ = ν . Then ther e is a m ap y 7→ µ y that asso ciates to every y ∈ Y a pr ob ability me asur e µ y on X su ch that i) F or every Bor el s ubset A ⊂ X , the function y 7→ µ y ( A ) is Bor el. ii) F o r ν -a.e. y ∈ Y , µ y ( π − 1 ( y )) = 1 . iii) µ = Z Y µ y dν ( y ) . 2.2. Discrete m easured group oids. The standard reference for measurable gr oup oids is [ 2 ]. The sour c e and r ange maps of a group oid ar e denoted by s a nd r , resp ectively . W e use sup erscr ipt 0 to denote the un it sp ac e of a group oid, as in G 0 for the g roup oid G . Our conv en tion is that a sub gr oup oid of a g r oup oid has the same unit space. A di scr ete me asur able gr oup oid G is a groupo id G equipp ed with the structure o f a stan- dard Borel space such that G 0 ⊂ G is a Bo rel subset, all the s tr ucture maps ar e Borel, and s − 1 ( { x } ) is coun table for a ll x ∈ G 0 . Let c s x and c r x denote the coun ting measures on s − 1 ( x ) and r − 1 ( x ), resp ectively . A di scr ete me asur e d gr oup oid ( G , µ ) is a discrete mea surable group oid G tog ether w ith a Borel measure µ on G 0 such that Z G 0 c s x dµ ( x ) = Z G 0 c r x dµ ( x ) A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 7 as measures on G . T his mea sure on G , which extends the one on G 0 , is also denoted by µ . Moreov er, ( G , µ ) is calle d inﬁn it e if s − 1 ( x ) is inﬁnite for all x ∈ G 0 in a subset o f p ositive measure. The discr ete mea sured gr oup oid ( G , µ ) is sa id to be er go dic if one (th us, all) of the following equiv alen t conditions hold: i) Any function f : G 0 → R that is G -invariant (that is, s ◦ f = r ◦ f ) is µ - a.e. constant. ii) F or any Borel subset A ⊂ G 0 of p ositive measure, the so-called satur ation A G : = { x ∈ G 0 ; ∃ γ ∈ G : s ( γ ) ∈ A, r ( γ ) = x } has full measure. iii) F or an y tw o Borel subsets A, B ⊂ G 0 of p ositive measure there ex is ts a Borel subse t E ⊂ G such that s ( E ) ⊂ A , r ( E ) ⊂ B a nd µ ( s ( E )) > 0, µ ( r ( E )) > 0. Each discrete mea sured group o id ( G , µ ) has a n er go dic de c omp osition , tha t is, there is an disintegration map ( G 0 , µ ) → ( X, ν ) such that ( G , µ x ) is for ν - a.e. x ∈ X er g o dic [ 16 , Theorem 6.1]. The measure space ( X, ν ) is called the sp ac e of er go dic c omp onents . A Bo rel map φ : ( G , µ ) → ( G ′ , ν ) is called a homomorphism if φ is a map o f gro up oids and f ∗ µ = ν . Let ( X, µ ) be a pr o bability spa ce with a probabilit y measure µ , and let Γ × X → X b e a measure preser ving (m.p.) group action. W e denote by ( X ⋊ Γ , µ ) the tr a nslation g r oup oid, i.e. the gro upo id with total space X × Γ, bas e space X a nd where s = π X and r : X ⋊ Γ → X is deﬁned to be the action of Γ on X . F o r the deﬁnition of an amenable or, equiv alen tly , hyp erﬁnite discrete measured group oid we refer to [ 26 , Chapter XII I, § 3]. 2.3. Sp ectral s equences. W eib el’s b o ok [ 29 ] is a s tandard refer ence for all the homolog ical algebra that we need. W e restate [ 29 , The o rem 5.8 .3 on p. 1 58] for the conv enience of the reader. Theorem 2. 3 (Grothendieck) . L et A , B and C b e ab eli an c ate gories, such that b oth A and B have enough inje ctive obje cts. L et G : A → B and F : B → C b e left exact functors, such that that G sends inje ctives to inje ctives. Then, given A ∈ A , ther e exists a ﬁrst quadr ant sp e ctr al se quenc e: E pq 2 = ( R p F )( R q G )( A ) = ⇒ R p + q ( F G )( A ) . Here R p F , for p ≥ 0, denotes the p -th rig h t derived functor of F . This v ery general sp ectral sequence can b e used to construct the classica l Ho chschild-Serre spectral sequence in group coho mology . The core o f our work consists in constructing suitable functors F a nd G and v erifying the abov e hypo thesis in the setting of discrete measured group oids . In view of the ab ove assumptions, it is useful to have a criterion that ensure s that a functor preserves injective ob jects: Lemma 2.4 ([ 29 , Theo r em 2.6.1 on p. 50]) . L et G : A → B b e a right exact functor b etwe en ab elian c ate gori es. If G has an exact left-adjoint functor, then it sends inje ctive s to inje ctive s. 2.4. L 2 -Betti num bers. The standard reference for L 2 -Betti num b e r s is L ¨ uck’s b o ok [ 21 ]. In this pap er we only refer to the ho mo logical-a lgebra- type deﬁnition of L 2 -Betti num b e rs in the context o f groups [ 21 , Chapter 6] and discrete mea sured g roup oids [ 23 ] (see also [ 27 ]). Let ( M , τ ) b e a ﬁnite von Neumann alge bra with a ﬁxed trace τ : M → C . L ¨ uc k int ro- duced a dimensio n function L 7→ dim ( M ,τ ) ( L ) for an arbitra r y M -module L with very nice 8 R OMAN S AUER AND ANDREAS THOM prop erties like additivit y for a rbitrary mo dules . If the co nt ext is clea r, we may als o omit the trace τ or the whole subscript in dim ( M ,τ ) . If M = L (Γ) is the g roup von Neumann algebra of a group Γ with its standar d tra ce, then the p -th L 2 -Betti n umber of Γ is deﬁned as b (2) p (Γ) : = dim L (Γ) T o r C Γ p  C , L (Γ)  = dim L (Γ) H p  Γ , L (Γ)  ∈ [0 , ∞ ] . If G is a discrete measured groupo id, then o ne deﬁnes in a similar wa y b (2) p ( G ) : = dim L ( G ) T o r R ( G ) p  L ∞ ( G 0 ) , L ( G )  ∈ [0 , ∞ ] . F o r the deﬁnition of R ( G ) and L ( G ) see Section 4.1 . 2.5. Op erators aﬃl iated wi th a ﬁnite v on Neumann algebra. Let ( M , τ ) again b e a ﬁnite von Neumann algebr a with a ﬁxed tr ace τ . Denote by L 2 ( M , τ ) the GNS-construction with resp ect to τ . There is a natura l a lgebra U ( M , τ ) o f closable, densely deﬁned and un b ounded op erator s on L 2 ( M , τ ), which are aﬃliated with the algebra M . F or details, w e refer to [ 25 , Chapter IX]. There is also the notion of dimension dim U ( M ,τ ) ( M ) ∈ [0 , ∞ ] for an a rbitrary U ( M , τ )- mo dule M [ 21 , Chapter 8; 22 ]. As it turns out, a us eful dimension function is o btained by restricting the mo dule structure to M and taking the dimension of the U ( M , τ )-mo dule as an M -mo dule. Again, we may omit τ or the whole subscript in dim U ( M ,τ ) if the context is clear. This a lgebra has b een studied in connectio n with L 2 -inv ariants in [ 22 , 28 ]. It sha r es very nice ring-theore tic prop erties, whic h w e w an t to summar ize in the s equel: i) U ( M , τ ) is von Neumann r e gular , i.e. all modules a r e ﬂa t, ii) U ( M , τ ) is left and right self-inje ct ive , i.e. U ( M , τ ) is injectiv e as left and as r ight mo dule ov er itself, iii) ι : M → U ( M , τ ) is a ﬂa t ring extension, i.e. L 7→ U ( M , τ ) ⊗ M L is exact, iv) dim U ( M ,τ ) U ( M , τ ) ⊗ M L = dim ( M ,τ ) L , for ev ery M -module L , v) dim U ( M ,τ ) hom U ( M ,τ ) ( L, U ( M , τ )) = dim U ( M ,τ ) L , for ev ery U ( M , τ )-mo dule L . vi) If dim U ( M ,τ ) L = 0, then hom U ( M ,τ ) ( L, U ( M , τ )) = 0. F o r pro o fs, w e r efer to [ 22 , 28 ] and the references therein. 2.6. Lo calization and completion. Let ( M , τ ) b e as ab ov e. Let A b e an ab elian catego ry with a faithful functor F : A → Mod( M ). Assume that the functor F pres e r ves limits and co-limits. In the categor y of M - mo dules, the sub-categ ory of zero-dimensio nal modules is a Serre sub-categor y . Moreover, the full sub- category S ⊂ A , given by those mo dules, that map to zero-dimensional mo dules, forms a Ser re sub-category as well. Example 2.5 . The example to hav e in mind, is the category of L ∞ ( X ) ⋊ Γ-mo dules with its forgetful functor to L ∞ ( X )-mo dules . Some of our co mputations will b e carr ied out in the lo c alize d category A / S , which we also denote by A lo c . This category is ab elia n and its prop erties, for the examples of special int erest, were studied in [ 27 ], wher e the seco nd author showed that it has eno ugh pro jective ob jects and natura lly embeds in A a s the s ub- category of those mo dules, which a r e c omple te with res p ect to the ra nk metric, see [ 27 ] for details. W e recall some of the results in Sectio n 4 . A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 9 3. Quo tients and Normal ity In this section, we provide the technical under pinning of the concept of (stro ng) extens io n of discrete measurable group o ids. 3.1. Ergo dic discrete me asurable g roup oids. Lemma 3 .1. L et ( G , µ ) b e an er go dic discr et e me asur e d gr oup oid with atom-fr e e µ , that is, µ ( { x } ) = 0 for every x ∈ G 0 . L et A, B ⊂ G 0 b e Bor el subsets of p ositive me asur e. Then µ  s − 1 ( A ) ∩ r − 1 ( B )  = ∞ . Pr o of. W e start with a general obser v atio n ab out er go dic gro upo ids: The function G 0 → Z ∪ {∞} , x 7→ # r − 1 ( x ) = # s − 1 ( x ) is a.e. constant b ecaus e o f ergodicity . If G 0 is atom-free, then # s − 1 ( x ) = ∞ for a.e. x ∈ G 0 since other wise there exis ted n ≥ 1 and A ⊂ G 0 with µ ( A ) < 1 / n and # s − 1 ( x ) ≤ n for x ∈ A , implying µ ( A G ) < 1 . Again by ergo dicity , w e can pick a set E ⊂ G with A ′ : = s ( E ) ⊂ A and B ′ : = r ( E ) ⊂ B such that µ ( A ′ ) , µ ( B ′ ) > 0. By the selection theore m we can a s sume that there is s uch E with injective s | E and r | E . Denote b y f : B ′ → E the in verse of r : E → B ′ . Cons ider the Borel map φ : s − 1 ( A ′ ) ∩ r − 1 ( B ′ ) → s − 1 ( A ′ ) ∩ r − 1 ( A ′ ) = G A ′ , φ ( γ ) = f ( r ( γ )) − 1 ◦ γ . Notice that the r estricted group oid G A ′ : = s − 1 ( A ′ ) ∩ r − 1 ( A ′ ) is ergo dic if G is so. Since φ is a ma p ov er A ′ with resp ect to the s ource maps and ﬁb erwise bijective, we obtain, with the g eneral observ ation ab ov e, that µ  s − 1 ( A ) ∩ r − 1 ( B )  ≥ µ  s − 1 ( A ′ ) ∩ r − 1 ( B ′ )  = µ  G A ′  = ∞ .  The following lemma is certainly w ell known but we failed to ﬁnd a reference. Lemma 3 .2. L et ( G , µ ) b e an er go dic discr ete me asur e d gr oup oid . Then ther e is a c oun table set I and a me asur e isomorphism φ : G 0 × I → G such that s ◦ φ = pr 0 G , wh er e pr 0 G : G 0 × I → G 0 is the pr oje ction and G 0 × I is endowe d with the pr o duct of µ and the c ounting me a sur e on I . F u rther, for every i ∈ I , the map G 0 → G 0 , x 7→ r ( φ ( x, i )) , is a me asure isomorphism. Pr o of. By er g o dicity , ( G 0 , µ ) is either discr ete (thus G is ﬁnite) or atom-free. W e leav e the easy pro of of the ﬁrst case to the reader and pro c e ed to the atom- free case . The following auxiliary fact is needed fo r the pro of. Claim: Let F ⊂ G b e a B o rel subset of ﬁnite measur e. Let E ⊂ G \ F b e a Borel subset on which r and s ar e injective, then there is a Borel subset D ⊂ G \ F containing E such that r , s a r e injective on D a nd s ( D ) = G 0 up to n ull se ts . Note that by G - inv a riance of µ , this also implies that r ( D ) = G 0 up to n ull sets. The s et B of Bor el sets D s uch that E ⊂ D ⊂ G \ F and r | D , s | D are injective c o ntains E and is partially ordered as follows: D 1 ≤ D 2 if and only if D 1 ⊂ D 2 up to null sets. Let T ⊂ B be a totally ordered subset. T o apply Zorn’s lemma la ter on, w e sho w that T has an upper bo und in B . Let r = sup D ∈T µ ( s ( D )) ∈ (0 , 1 ]. Pick a coun table family { D n } n ∈ N in T with D n ≤ D n +1 and µ ( s ( D n )) → r . Let D = S D n . Upo n subtra cting a suitable n ull s e t from D we can ensur e that fo r all x, y ∈ D there is n 0 ∈ N with x, y ∈ D n 0 . Thus r | D , s | D are injectiv e, and D is an upp er bo und of T . By Zorn’s lemma the set B p ossesses a maximal element D max ∈ B . Supp ose µ ( s ( D max )) < 1, thus µ ( s ( D max )) < 1. Let A ⊂ G 0 \ s ( D max ) and B ⊂ G 0 \ r ( D max ) b e subsets of p ositive mea sure. Since µ ( F ) < ∞ b y assumption, 10 R OMAN S AUER AND ANDREAS THOM Lemma 3.1 implies that µ ( s − 1 ( A ) ∩ r − 1 ( B ) ∩ ( G \ F )) = ∞ . In pa rticular, there is E ⊂ G of po sitive measure such that s ( E ) ⊂ A and r ( E ) ⊂ B . O nce more using the se le ction theor em, we can assume that there is such E with r, s b eing injective on E . Since E ∪ D max would contradict maximality of D max , it follo ws that µ ( s ( D max )) = µ ( r ( D max )) = 1 . Let us con tin ue with the pro of o f the lemma. Using the selection theorem, pic k a coun t- able Bor el partition { E n } n ∈ N of G such that r, s a re injective on each E n . W e construct inductively B o rel subsets D n ⊂ G such that i) E n ⊂ S n i =1 D i for ev ery n ≥ 1, and the union is disjoint up to null sets, ii) r and s are injectiv e on D n for ev ery n ≥ 1, iii) r ( D n ) = s ( D n ) = G 0 up to n ull sets. According to the c laim ab ov e there is such D 1 , and fo r n > 1 there is a Bo rel subset D n ⊂ G \ S n − 1 i =1 D i containing E n \ S n − 1 i =1 D i such that r , s are injective on D n and s ( D n ) = G 0 up to null se ts. Since E n ⊂ S i ≥ 1 D i for ev ery n ∈ N , we have G = S i ≥ 1 D i . The (in verse of the) desired isomorphism is now giv en as G → G 0 × N , γ 7→ ( s ( γ ) , n ) if γ ∈ D n .  Deﬁnition 3.3. Let S ⊂ G be a subgroup oid o f a discr ete measur ed g roup oid G . Le t A ⊂ G 0 be a Bo rel subset. Let End S ( G ) | A denote the set of all Borel maps φ : A → G such that i) s ( φ ( x )) = x for a.e. x ∈ A , ii) γ ∈ S ⇔ φ ( r ( γ )) γ φ ( s ( γ )) − 1 ∈ S for ev ery γ ∈ G with r ( γ ) ∈ A and s ( γ ) ∈ A . Let Aut S ( G ) | A ⊂ End S ( G ) | A be the subset consisting of those φ such tha t x → r ( φ ( x )) is a.e. injective. An element in Aut S ( G ) | A is called a lo c a l se ction of G . If A = G 0 , we will drop the subscr ipt A in the notatio n. If S = G , the condition ii) is void, and we will dro p the subscript S in the notation. Deﬁnition 3.4 . L e t φ : A → G a nd ψ : B → G b e lo cal sections. The c omp ositio n φ ◦ ψ is the loc a l section deﬁned by ψ − 1  r ( B ) ∩ A  → G , x 7→ φ  r ( ψ ( x ))  ◦ ψ ( x ) . 3.2. Deﬁnition of s trong extensions . The following notion is a genera lization of (stro ng) normality for equiv alence relations [ 9 , Theorem 2.2 and Deﬁnition 2 .14]. Deﬁnition 3.5. W e call a subgroup oid S ⊂ G o f a discr ete mea sured group oid (str ongly) normal if there is a countable family { φ n } (in Aut S ( G )) in End S ( G ) such that for a .e . γ ∈ G there exists exactly one φ n in the family with φ n ( r ( γ )) γ ∈ S . W e say that { φ n } is a family of (str ongly) normal choic e functions . R emark 3 .6 . Note that for a translation group oid X ⋊ G and a subgroup H ⊂ G , the group oid X ⋊ H ⊂ X ⋊ G is strongly normal if and only if H is norma l in G . A coun table family of strongly no r mal choice functions is given b y the automorphisms o f X which are induced b y an y complete set o f representativ es of the cosets G/H . The previous deﬁnition makes sure that a ll the r elev a nt choices can b e made in a mea - surable wa y . Deﬁnition 3.7. Let θ : ( G , µ ) → ( Q , ν ) b e a ho mo morphism of discrete measured group oids. Let S = θ − 1 ( Q 0 ) b e the k ernel of θ , a nd let µ = R Q 0 ν y dν ( y ) b e the measure disint egratio n with resp ect to θ . Then A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 11 i) θ is a surje ction if for a .e. x ∈ G 0 and a .e. γ ∈ Q with s ( γ ) = θ ( x ) there exists g ∈ G with θ ( g ) = γ ( p ointwise su rje ct ivity ), ii) θ is a str ong surje ction if θ is a surjection and ( S | θ − 1 ( y ) , ν y ) is ergo dic for a .e. y ∈ Q 0 . The following lemma is s tandard and proved in the context of equiv alence relations in [ 11 , Lemma 2.1]. W e presen t a pro of of the group oid v ersion here so that the rea der can see the v a lidit y of Remark 3.9 . Lemma 3.8. L et G b e an er go dic discr ete me asur e d gr oup oi d. L et E , F ⊂ G 0 b e Bor el subsets with the same me asur e. Then ther e is φ ∈ Aut( G ) such that r ( φ ( E )) ⊂ F and r ◦ φ : E → F is a me asur e isomorphism. Pr o of. Cho ose a meas ure iso morphism G 0 × I → G , ( x, i ) → φ i ( x ) as in Lemma 3.2 . F or any Borel subsets A, B ⊂ G 0 deﬁne c ( A, B ) : = sup n ∈ N µ  r ( φ n ( A )) ∩ B  . By ergo dicity , c ( A, B ) > 0 whenever µ ( A ) > 0 and µ ( B ) > 0 . Let E 0 = E and F 0 = F . By induction we deﬁne Bo rel sets E n , F n ⊂ G 0 and element s ψ n ∈ { φ i ; i ∈ N } a s follows: Given E n and F n let i ∈ N b e the minimal n umber suc h that µ ( r ( φ i ( E n )) ∩ F n ) ≥ c ( E n , F n ) / 2 , and let ψ n = φ i and E n +1 = E n \ ( r ◦ φ n ) − 1 ( F n ), F n +1 = F n \ ( r ◦ ψ n )( E n ). Let E ∞ = T n E n and F ∞ = T n F n . F or ev ery n ∈ N , the s ets E n and F n hav e the same meas ure, th us µ ( E ∞ ) = µ ( F ∞ ). F urther, we hav e µ ( E ∞ ) = µ ( F ∞ ) = 0 since otherwise c ( E n , F n ) ≥ c : = c ( E ∞ , F ∞ ) > 0 for all n which contradicts the choice of ψ n at the s ta ge where µ ( E n \ E n +1 ) < c/ 2. So ( E ′ n ) n ∈ N and ( F ′ n ) n ∈ N deﬁned b y E ′ n = E n \ E n +1 and F ′ n = F n \ F n +1 are partitions of E and F , res p ectively . No w w e can deﬁne φ : G 0 → G as follows: φ ( x ) = ( id x if x 6∈ E , ψ n ( x ) if x ∈ E ′ n .  R emark 3.9 . After ﬁxing a measure isomorphism G 0 × I → G , the co nstruction of φ ∈ Aut( G ) in the previo us lemma involv es no o ther choices. This fact is important in a situation where one w ants to conclude that φ dep ends on the input data µ, E , F in a measurable w ay . The next lemma turns o ut to b e crucia l for the cons truction of our spe ctral sequence in Section 4 . Lemma 3.10. L et θ : ( G , µ ) → ( Q , ν ) b e a st r ong surje ct ion of er go dic, discr ete me asur e d gr oup oids. F or every Bor el subset A ⊂ Q 0 and every φ ∈ Aut( Q ) | A ther e is a lift ψ ∈ Aut( G ) | θ − 1 ( A ) ∩G 0 , that is, θ ( ψ ( x )) = φ ( θ ( x )) for a.e. x ∈ θ − 1 ( A ) ∩ G 0 . Pr o of. Let S b e the k ernel of θ . If the cardina lit y of G 0 is ﬁnite, Lemma 3.10 becomes very easy and is left to the r eader. W e assume in the s e quel that G 0 is inﬁnite. Thus there is a Borel isomorphism G 0 ∼ = [0 , 1]. Let us ﬁx one. Let µ = R Q 0 µ y dν ( y ) b e the disint egratio n of ( G 0 , µ ) with respect to θ . By assumption, S | θ − 1 ( y ) is ergo dic for a.e. y ∈ Q 0 . First of all, w e r epro duce an argument in [ 15 , P ro of of Theorem 3.18 on p. 70]. The function p : x 7→ µ θ ( x ) ( { x } ) on G 0 is measurable and G -inv ar iant. By er go dicity we hav e p ( x ) = c for µ -a.e. x ∈ G 0 and a constant c ≥ 0. If c > 0, then θ : G 0 → Q 0 has ﬁnite ﬁber s ν -a.e., a nd the lemma is easily proved using the selection theor em. W e leave that to 12 R OMAN S AUER AND ANDREAS THOM the r e ader a nd re strict ourselves to the mo re complicated case that c = 0, that is, µ y is contin uous for ν -a .e . y ∈ Q 0 . Consider the follo wing Borel ma ps f : G 0 → Q 0 × [0 , 1] , f ( x ) = ( θ ( x ) , µ θ ( x ) ([0 , x ])) , g : Q 0 × [0 , 1] → G 0 , ( y , t ) 7→ min { x ∈ G 0 ; µ y ([0 , x ]) ≥ t } . One sees that g ◦ f = id. In pa rticular, f is injective. F urther we have for a Borel subset A ⊂ Q 0 µ  f − 1 ( A × [0 , t ])  = Z A µ y  g ( A × [0 , t ])  dν ( y ) = Z A µ y  g ( { y } × [0 , t ])  dν ( y ) = Z A µ y  [0 , g ( y , t )]  dν ( y ) = t · ν ( A ) . Thu s f ∗ µ = ν × λ , where λ denotes the Lebes g ue measur e. Hence im( f ) has full mea sure, and f is a measure isomorphism ( G 0 , µ ) → ( Q 0 × [0 , 1] , ν × λ ). Since θ is point wise surjective, there is a, at lea st set-theoretic, lift ψ ′ of φ , a nd one use s Lemma 3.2 to obtain a measurable suc h ψ ′ ∈ End( G ) | θ − 1 ( A ) ∩G 0 . By the selection theore m there is a partition θ − 1 ( A ) ∩ G 0 = [ k ≥ 1 D k with ψ ′ | D k ∈ Aut( G ) | D k for an y k ≥ 1. Let D ′ k : = r ( ψ ′ ( D k )). In the following, we neglect null sets. By the uniqueness of disinte- gration w e obtain that µ r ( φ ( y )) ( D ′ k ) = µ y ( D k ) for ev ery y ∈ Q 0 . The pr o blem is that the sets { D ′ k } k ≥ 1 could have o verlaps, th us causing ψ to b e non- injectiv e. The idea is to make these sets ﬁberwis e disjoint b y an applicatio n of Lemma 3.8 to ( S , µ y ) for y ∈ C . T o make this precise, cons ider for y ∈ Q 0 the unique minimal sequence 0 = m ( y , 0) < m ( y , 1) < m ( y , 2) < . . . of n um b ers in [0 , 1] with the property µ r ( φ ( y )) ( D ′ k ) = µ r ( φ ( y ))  g ( Q 0 × [ m ( y, k − 1) , m ( y , k )])  . Let τ y k : D ′ k → S b e the lo cal s e c tion o f ( S , µ r ( φ ( y )) ) co nstructed in Lemma 3.8 with the prop erty that r ◦ τ y k : D ′ k → g  Q 0 × [ m ( y, k − 1) , m ( y , k )]  is a µ r ( φ ( y )) -measure isomorphism. W e deﬁne a local section ψ on θ − 1 ( A ) b y ψ ( x ) = τ θ ( x ) k ◦ ψ ′ ( x ) for x ∈ D k . It follows from the explicit co nstruction of τ y k (see Remark 3.9 ) that ψ is mea s urable. It is clear that ψ is still a lift of φ since τ θ ( x ) k is (ﬁb erwise ) a lo ca l section of S = ker( θ ). One easily veriﬁes that µ ( r ( ψ ( B ))) = µ ( B ) for a ll Bore l subsets B ⊂ θ − 1 ( A ), th us ψ : θ − 1 ( A ) → G lies indeed in Aut( G ) | θ − 1 ( A ) ∩G 0 .  A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 13 3.3. Strongl y normal subgroup oids vs. strong quotie n ts. Theorem 3.11. L et θ : G → Q b e a st r ong su rje ct ion of erg o di c, discr et e me asur e d gr oup oids. Then the kernel ker( θ ) = θ − 1 ( Q 0 ) is a str ongly normal sub gr o up oid of G . Pr o of. Consider a measure isomorphism φ : Q 0 × I → Q as in Theore m 3.2 . Acco rding to Lemma 3 .10 we can lift each φ i : = φ | Q 0 ×{ i } ∈ Aut( Q ) to ψ i ∈ Aut( G ). Then { ψ i } i ∈ I is a family of strongly no rmal c hoice functions for ker( θ ).  The following theo rem is a straightforw ard generalization of [ 9 , Theo rem 2.2], where S and G ar e assumed to b e e q uiv a le nce relatio ns, to group oids. One do es the same co nstructions as in lo c. cit. line b y line, but fo r gr oup oids. Theorem 3.12 (Quotient constr uction) . L et S ⊂ G b e a str ongly normal sub gr oup oid of an er go dic discr ete me asur e d gr o up oid . Then ther e is a str ong sur je ction θ : G → Q onto an er go dic discr ete me asur e d gr oup oid, c al le d the q uotient of G by S , su ch that i) ker( θ ) : = θ − 1 ( Q 0 ) = S , ii) for a.e. x ∈ G 0 and every γ ∈ Q with θ ( x ) = s ( γ ) t her e exists g ∈ G s uch θ ( g ) = γ , iii) for any er go dic discr ete me asur e d gr oup oid Q ′ and any homomorp hism θ ′ : G → Q ′ with S ⊂ ker θ ′ ther e is a m.p. homomorp hism κ : Q → Q ′ such t hat κ ◦ θ = θ ′ . Deﬁnition 3.1 3. With the setting of the previous theorem, we say that ( G , µ ) is a st r ong extension of ( S , µ ) by ( Q , ν ), and we indicate this, similarly to gro ups, by writing: 1 → ( S , µ ) − → ( G , µ ) − → ( Q , ν ) → 1 W e rec o rd the following lemma for later refer ence. Lemma 3. 14. We r etain the notation of the pr e c e di ng The or em 3. 12 . F or every lo c al se ction ψ : A → G t her e is a c ountable Bor el p artition A = S n ≥ 1 A n , and lo c al se ctions q n ∈ Aut S ( G ) | A n and s n ∈ Aut( S ) such that ψ | A n = q n ◦ s n Pr o of. Let { φ n } n ≥ 1 be a family of strongly normal c hoice functions. Deﬁne A n : = { a ∈ A ; ( φ n ◦ ψ )( a ) ∈ S } . Then s n : = φ n ◦ ψ | A n ∈ End( S ) | A n . Upon further par titioning each A n , we can assume that s n ∈ Aut( S ). F or q n = φ − 1 n the assertion follo ws.  R emark 3.1 5 . Let S ⊂ G b e stro ngly normal with quotien t G θ − → Q a s in Deﬁnit ion 3.13 . A lo cal sectio n φ ∈ Aut S ( G ) is the lift of so me φ ′ ∈ Aut( Q ) b y the following ar gument: The map θ ◦ φ : G 0 → Q is S -in v aria nt . The restr ictions of S to ﬁb ers θ − 1 ( q ) for q ∈ Q 0 are ergo dic since θ is a strong surjection. Thus, θ ◦ φ descends to a map Q 0 → Q of which φ is a lift. R emark 3.16 . Consider Gab oria u’s extension ( 5.1 ) of Subsection 5.2 1 → ( Z ⋊ Λ , µ × ν ) − → ( Z ⋊ Γ , µ × ν ) − → ( Y ⋊ Q, ν ) → 1 asso ciated to a gr oup extension 1 → Λ → Γ p − → Q → 1. If the action of Γ on Z = X × Y is not ergo dic, then this is not a str ong extension in the sense of Deﬁnition 3.13 . How ev er, the conclusion from the combination of Lemma 3.14 and Remark 3.15 is still true: F or every lo cal se c tion ψ : A → Z ⋊ Γ there is a countable Borel partition A = S n ≥ 1 A n , lo cal sections q n : A n → Z ⋊ Γ that ar e lifts o f lo cal sections of Y ⋊ Q , a nd lo cal sections s n : A n → Z ⋊ Λ such that ψ | A n = q n ◦ s n . 14 R OMAN S AUER AND ANDREAS THOM F o r that, note that a lo cal section ψ : A → Z ⋊ Γ is es sentially given by a map A → Γ that we denote b y the same name. Ther e a re a coun table Bo r el partition Y = S n ≥ 1 Y n and elements q n ∈ Q for every n ≥ 1 such that p ( ψ ( a )) = q n if a ∈ A n = A ∩ X × Y n . Choose lifts γ n ∈ Γ for ea ch q n ∈ Q . Then the desired q n : A n → Γ and s n : A n → Λ are deﬁned by q n ( a ) = γ n and s n ( a ) = ψ ( a ) q n ( a ) − 1 . 4. Construction of the spectral sequence 4.1. R ( G ) -m o dules . Let ( G , µ ) be a discrete mea sured group oid with s ource and r ange maps s, r : G → G 0 . Denote b y L ( G , µ ) the assoc ia ted von Neumann algebra and by U ( G , µ ) the algebra of o p e r ators, which are aﬃliated with the ﬁnite von Neumann algebra L ( G , µ ). Recall that a lo cal s ection of ( G , µ ) is a Bo rel sec tion φ : A → G of s such that the r e s triction of r to φ ( A ) is injectiv e. Let R ( G , µ ) b e the conv olution a lgebra o f c omplex v alued Borel functions on G , which can be written a s ﬁnite sums of pr o ducts of es s ent ially b ounded complex - v a lued B o rel functions on G 0 and c haracteris tic functions χ φ ( A ) of g raphs o f lo ca l sections φ : A → G . Clear ly , R ( G , µ ) ⊂ L ( G , µ ) is a sub-ring. R emark 4 .1 . W e consider the case ( G , µ ) = ( X ⋊ Γ , µ ) of a translation group oid. The cr osse d pr o du ct ring L ∞ ( X ) ⋊ Γ is the free L ∞ ( X )-mo dule with basis Γ. Its m ultiplication is determined by the rule γ f = f ( γ − 1 ) γ for γ ∈ Γ , f ∈ L ∞ ( X ). The crossed pr o duct ring em b eds as a r ank-dense subring in to R ( X ⋊ Γ , µ ) via P f g g 7→ f ∈ L ∞ ( X ⋊ Γ) with f ( x, g − 1 ) = f g ( x ) (compare [ 27 , Propo sition 4.1]). The following lemma is a useful c haracter ization of mo dules ov er R ( G , µ ), which should be se e n in a nalogy to group rings. Lemma 4.2 . Le t M b e an ab eli an gr oup. T o give a R ( G , µ ) -mo dule structur e on M is t he same as to give a L ∞ ( G 0 ) -mo dule struct ur e and c omp atible p artial isomorphism as fol lows: F or e ach lo c al se ction φ of ( G , µ ) , ther e exists an isomorphism ˆ φ : χ A M → χ r ( φ ( A )) M that is c omp atible with c omp osition, r estriction and ortho gonal sum. Pr o of. Denote by R ( G , µ ) the universal L ∞ ( G 0 )-algebra , generated by symbols φ for every lo cal section φ of G , sub ject to relatio ns implemen ting compatibility with restric tio n, com- po sition and orthog o nal sum. Clearly , the natura l map σ : R ( G , µ ) → R ( G , µ ) is surjective and M is a R ( G , µ )-mo dule. The proo f is ﬁnis hed b y showing that σ is injectiv e and hence an isomorphism. Let h = P f i φ i be a ﬁnite sum in R ( G , µ ). It follows fr om the compatibilit y with resp ect to comp ositio n and restr iction that e very element in R ( G , µ ) can b e wr itten in this form. Assume that P f i φ i = 0. In order to show injectivity , w e have to prove h = 0. W e may assume that the supp ort of f i is e qual to the domain of φ i . If we partition G 0 int o sets X 1 , . . . , X k , on whic h (i) the domains of the lo cal sections φ i | X j are either X j or empt y , a nd (ii) the lo cal sections φ i | X j are either a.e. equal or a.e. diﬀerent, we can res trict our attention to o ne set X j at a time. Indeed, since φ i is the orthog onal sum o f its res trictions to the X j , it suﬃces to s how P i f i | X j φ i | X j = 0. Without los s of generality , we can now assume that the lo cal sections φ i | X j are a .e. diﬀeren t from each A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 15 other. Clear ly , P i f i | X j φ i | X j = 0 is only p ossible if f i = 0, since the φ i | X j hav e disjoint suppo rt as c haracteristic functions on G . This ﬁnishes the pro of.  R emark 4.3 . The ab elian group L ∞ ( G 0 ) beco mes a n R ( G , µ )-mo dule v ia ˆ φ : L ∞ ( A ) → L ∞ ( r ( φ ( A ))) , f 7→  x 7→ f (( r ◦ φ ) − 1 ( x ))  for a local section φ : A → G . 4.2. Com pletion and lo calization of mo dules. E very L ∞ ( G 0 )-mo dule M carries a canonical metric ( r ank metric ), whic h is induced by the so-called r ank , i.e. d ( ξ , η ) = inf { µ ( A c ) | A ⊂ G 0 Borel , χ A ξ = χ A η } , ∀ ξ , η ∈ M . It w as shown in [ 27 , Lemma 4.4 ] that the completion of a R ( G , µ )-mo dule with resp ect to the underlying L ∞ ( G 0 )-mo dule carr ies a natural R ( G , µ )-mo dule structure and that the asso ciated completion functor is ex act [ 27 , Lemma 2.6]. Mor eov er, the ca teg ory of complete R ( G , µ )-mo dules was shown to b e ab elian with enough pro jective ob jects [ 2 7 , Theorem 2.7]. W e denote by Mo d( R ( G , µ )) comp the full sub ca tegory of Mo d( R ( G , µ )), formed b y com- plete R ( G , µ )-mo dules and denote b y c : Mo d( R ( G , µ )) → Mo d( R ( G , µ )) comp the completion functor. Pro jectives in Mo d( R ( G , µ )) comp are obtained by co mpleting free mo dules. The following lemma sho ws that there a re also enough injective ob jects. Lemma 4.4. The ab elian c ate go ry Mo d( R ( G , µ )) comp has enou gh inje ctive obje cts. Pr o of. Clearly , the ab elian category Mo d( R ( G , µ ) ) has enoug h injective ob jects. Let Z b e an injective ob ject in the a b elian categ ory Mo d( R ( G , µ )). It is s uﬃcient to construct an complete injective ob ject Z ′ , which contains Z a s a submo dule. Let α b e a n o rdinal with uncountable coﬁnality , i.e. ther e exists no countable co ﬁna l subset. F or every β ≤ α , we deﬁne Z β by the following tr ansﬁnite inductive pro cedure. W e se t Z 0 = Z and Z β +1 to be some injective R ( G , µ ) -mo dule, containing the completion of Z β as a sub-mo dule. If β is a limit ordinal, w e s et Z β = ∪ β ′ <β Z β ′ . Every Cauch y se quence in Z α is contained in some Z β for β < α and th us its limit exis ts in Z β +1 ⊂ Z α . Hence Z α is complete. Mor eov er, cho osing α big enough, Z α is injective as a R ( G , µ ) -mo dule, since injectivity can be tested on the set of sub-modules of the trivial module. W e can deﬁne Z ′ = Z α .  R ( G , µ )-mo dules, which are zer o-dimensional as L ∞ ( G 0 )-mo dules, complete to the zero mo dule. Indeed, this is a reformulation of the lo cal criterion of zer o-dimensionality , that can b e fo und in [ 23 , Theorem 2 .4]. Hence, there exists a natura l e x act functor of ab elia n categorie s (4.1) c : Mo d( R ( G , µ )) lo c → Mo d( R ( G , µ )) comp , which is a n equiv alence o f a belia n ca tegories . Indeed, the inv erse is the res triction of the natural quo tient functor. In the sequel we concentrate on complete R ( G , µ )-mo dules. The completion of L ∞ ( G 0 ) identiﬁes natura lly with the algebra of mea surable functions on G 0 , which we denote b y M ( G 0 ). Note that U ( G , µ ) is complete, wher eas L ( G , µ ) is not necessarily . 16 R OMAN S AUER AND ANDREAS THOM 4.3. Some deriv ed functors. In this subsection, we intro duce s ome der ived functors, which are appro priate to make our a pproach to a spec tral sequence work. They are de- ﬁned on the lo calized categor y; and thus we a r e heavily using its nice homolog ical prop erties which were e stablished in preceding section a nd [ 27 ]. Deﬁnition 4.5 . Let M b e a complete left R ( G , µ )-mo dule. W e deﬁne: H ∗ ( G , M ) = T or R ( G ,µ ) ∗ ( M ( G o ) , M ) , and H ∗ ( G , M ) = E xt ∗ R ( G ,µ ) ( M ( G 0 ) , M ) , to be the c omp lete G -ho mology and G -cohomo logy of the R ( G , µ )-mo dule M . Here, T or R ( G ,µ ) ∗ ( M ( G 0 ) , ?) and E xt ∗ R ( G ,µ ) ( M ( G 0 ) , ?) denote the der ived functor s of the functors M 7→ M ( G o ) ⊗ R ( G ,µ ) M , and M 7→ hom R ( G ,µ ) ( M ( G 0 ) , M ) , from the ab elia n categor y Mo d( R ( G , µ )) comp to ab elian gr o ups. F ollowing the arguments in [ 29 , Chapter 2.7], we see that the bi-functors T or R ( G ,µ ) ∗ (? , ?) and Ext ∗ R ( G ,µ ) (? , ?) are balanced, as are their classical co un terparts. 4.4. Sp ectral sequence. The following theorem enables us to construct a sp ectral sequence for the cohomology which w e just deﬁned. Theorem 4.6. L et 1 → ( S , µ ′ ) → ( G , µ ) → ( Q , ν ) → 1 b e a stro ng extens ion of discr ete me asur e d gr oup oi ds, i.e. ( G , µ ) → ( Q , ν ) is a str ong sur- je ct ion with kernel ( S , µ ′ ) . L et M b e a c omple te left-mo dule over R ( G , µ ) . The ab eli an gr oup hom R ( S , µ ′ ) ( M ( G 0 ) , M ) is natur al ly a c omplete left mo dule over R ( Q , ν ) , and ther e is a natur al isomo rphism of ab elian gr oups hom R ( Q ,ν )  M ( Q 0 ) , hom R ( S , µ ′ ) ( M ( G 0 ) , M )  = hom R ( G ,µ ) ( M ( G 0 ) , M ) . Pr o of. Since M ( Q 0 ) ⊂ M ( G 0 ), there is a natur al L ∞ ( Q 0 )-mo dule structure on hom R ( S , µ ′ ) ( M ( G 0 ) , M ) . It re ma ins to pr ovide partial isomo rphisms as des crib ed a b ove. Let φ b e a lo ca l section of Q . Since ( G , µ ) → ( Q , ν ) is a s trong surjection, there ex ists a s ection φ ′ of G whic h lifts φ b y Lemma 3.10 . F o r f ∈ hom R ( S , µ ′ ) ( M ( G 0 ) , M ), w e deﬁne ( φ ⊲ f )( g ) = φ ′ f ( φ ′− 1 g ) , ∀ g ∈ M ( G 0 ) . This is well-deﬁned and, together with the afor ementioned M ( Q 0 )-mo dule structure, it deﬁnes a left-mo dule structure of R ( Q , ν ). Moreov er, hom R ( S , µ ′ ) ( M ( G 0 ) , M ) is easily seen to be complete. W e now a im to sho w that hom R ( Q ,ν )  M ( Q 0 ) , hom R ( S , µ ′ ) ( M ( G 0 ) , M )  = hom R ( G ,µ ) ( M ( G 0 ) , M ) . An element in f ∈ hom R ( G ,µ ) ( M ( G 0 ) , M ) is descr ib ed b y the image of 1 ∈ M ( G 0 ). F or this, note that all L ∞ ( Q 0 )-mo dule maps are automatically contractive and hence c o ntin uous in the rank metr ic , see [ 27 , Lemma 2.3]. Let φ b e a lo c al section of ( G , µ ). W e set ˆ φ = χ φ ( A ) ∈ R ( G , µ ). The image of 1 under f satisﬁes ˆ φf (1) = f ( χ r ( φ ( A )) ) = χ r ( φ ( A )) f (1) . A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 17 Conv ersely , every suc h elemen t giv es r ise to a mo dule homomor phism. An elemen t in hom R ( S , µ ′ ) ( M ( G 0 ) , M ) is describ ed as an element in M , satisfying the ab ov e rela tion for local sections of ( S , µ ′ ). Having th is description, it is obvious that hom R ( Q ,ν )  M ( Q 0 ) , hom R ( S , µ ′ ) ( M ( G 0 ) , M )  ident iﬁes with hom R ( G ,µ ) ( M ( G 0 ) , M ) s ince we ca n wr ite ea ch lo cal sec tion o f ( G , µ ) as a countable orthogonal sum of lo ca l sections, which are pro ducts of lifts of a lo cal section of ( Q , ν ) and a lo cal section of ( S , µ ′ ) by Lemma 3.14 and Remark 3.15 . This ﬁnishes the pro of, since M is complete and the ortho g onal decomp osition gives rise to a rank conv ergent sum M .  The decomp osition of functors which was es tablished in the preceding theorem yields a Grothendieck sp ectral sequence by Theo rem 2.3 : Theorem 4.7. L et 1 → ( S , µ ′ ) → ( G , µ ) → ( Q , ν ) → 1 b e a str ong extension of discr ete me asur e d gr oup oids. L et M b e a c omplete left R ( G , µ ) - mo dule. Then ther e is a ﬁrst quadr ant sp e ct ra l se quenc e with E 2 -term E pq 2 = Ext p R ( Q ,ν )  M ( Q 0 ) , Ext q R ( S , µ ′ )  M ( G 0 ) , M   (4.2) = H p ( Q , H q ( S , M )) c onver ging to Ext p + q R ( G ,µ )  M ( G 0 ) , M  = H p + q ( G , M ) . F or M = U ( G , µ ) , which is a R ( G , µ ) - U ( G , µ ) -bimo dule, we obtain a sp e ctr al se quenc e of U ( G , µ ) -mo dules. Pr o of. The only claim tha t r emains to b e veriﬁed is that hom R ( S , µ ′ ) ( M ( R 0 ) , ?) s ends in- jective o b jects to injectiv e ob jects. How ever, this follo ws from Lemma 2.4 if we can provide an exact left a djo int functor. Clearly , a similar argumen t as a bove shows that M 7→ c  M ( R 0 ) ⊗ M ( Q 0 ) M  is left adjo int to hom R ( S , µ ′ ) ( M ( R 0 ) , ?). Mo reov er, it is the comp osition of a ﬂa t ring extension (note that M ( Q 0 ) is von Neumann regular ) a nd the exa ct co mpletion functor, and th us exact. This ﬁnishes the proof.  R emark 4.8 . F o r later reference w e no te that the E 1 is of the spectral sequence is giv en b y E p,q 1 = hom R ( Q,ν )  P p , Ext q R ( S , µ ′ )  M ( G 0 ) , M   for a pro jectiv e r e s olution P ∗ of M ( Q 0 ) in Mo d( R ( Q , ν )) comp . R emark 4.9 . The conclusion of Theorem 4.7 also holds true for Gabo riau’s extension ( 5.1 ) even if the group Γ do es no t ac t ergo dically and so Gab or iau’s extension is not a strong extension in the sens e of Deﬁnition 3.13 : F or that, note that the co nclusion of Lemma 3.10 , which is used in Theore m 4.6 , is very easy to s ee for Gab oria u’s e x tension (compare the argument in Rema rk 3.1 6 ). F urthermor e, o ne repla ce the a pplication of Lemma 3.14 and Remark 3.15 in the pro o f of Theorem 4 .6 by Remark 3.16 . All other s teps in the pro ofs of Theorems 4.6 and 4.7 sta y the sa me in the ca se of Ga bo riau’s extension. 18 R OMAN S AUER AND ANDREAS THOM 4.5. Iden tiﬁcation o f L 2 -Betti num bers . In order to apply the spectr al sequence from Section 4 to the computation of L 2 -Betti num b e rs of gr oups, we need to study the sp ecial case of a m.p. gr oup action mo re clos ely . Let Γ b e a disc r ete gr o up, a nd let ( X , µ ) b e a standard Bor el pr obability spa ce on which Γ acts by m.p. Bor el is omorphisms. Note that we do not impo se an y conditions lik e ergodicity or freeness of the action. In many situations, in v ar iants of a m.p. action of a discre te group ar e actually in v aria nts of the group itself. In the sequel we wan t to iden tify the dimensions of the homological and cohomolog ical inv ariants for ( X ⋊ Γ , µ ), which we just in tro duced, with ordinary L 2 -Betti nu mbers of Γ. First of all, fo r any discrete mea sured group oid ( G , µ ), there is a natur al tr ansformation (4.3) T or R ( G ,µ ) ∗ (? , L ( G , µ )) → T or R ( G ,µ ) ∗ (? , U ( G , µ )) , consisting of dimension isomorphisms. Indeed, it is induced b y the natura l map ? ⊗ R ( G ,µ ) L ( G , µ ) → c (?) ⊗ R ( G ,µ ) U ( G , µ ) . F o r a pro of that this is a dimensio n iso morphism, we refer to the pro of of [ 27 , P r op osition 4.7]. If follows from [ 27 , Lemma 1.1] that the induced map o n derived functors are dimension isomorphisms as w ell. Secondly , there is a n isomorphism of r ight U ( G , µ )-mo dules hom U ( G , µ )  T o r R ( G ,µ ) ∗ (? , U ( G , µ )) , U ( G , µ )  ∼ = Ext ∗ R ( G ,µ ) (? , U ( G , µ )) , since U ( G , µ ) is self-injectiv e. Moreover, s ince dualizing is dimension preserving, w e g et: (4.4) dim T or R ( G ,µ ) ∗ (? , U ( G , µ )) = dim E x t ∗ R ( G ,µ ) (? , U ( G , µ )) . In [ 23 ] the ﬁrst author o bserved that due to dimension ﬂatness of the r ing extensions C Γ ⊂ R ( X ⋊ Γ) , and L (Γ) ⊂ L ( X ⋊ Γ) , there are natural dimension is o morphisms as follo ws: (4.5) T o r R ( G ,µ ) ∗ ( L ∞ ( G 0 ) , L ( G , µ )) ∼ = H ∗ (Γ , L ( X ⋊ Γ)) ∼ = H ∗ (Γ , L (Γ)) ⊗ L Γ L ( X ⋊ Γ) . Thu s, com bining the dimension isomorphisms ( 4.3 ) for L ∞ ( G 0 ), and ( 4.4 ), and ( 4.5 ), we get: b (2) ∗ (Γ) = dim L ( X ⋊ Γ) H ∗ ( X ⋊ Γ , L ( X ⋊ Γ)) (4.6) = dim L ( X ⋊ Γ) H ∗ ( X ⋊ Γ , U ( X ⋊ Γ)) R emark 4.10 . Note that H n ( G , U ( G , µ )) is the U ( G , µ )-dua l of H n ( G , U ( G , µ )) a nd th us has the pleasant feature that it v anishes as so on as its dimension is z e r o. This follows from the results in [ 28 ]. 5. E xamples of gr oupoid extensions 5.1. Group extensi ons. Let Γ b e a gro up and le t Λ ⊂ Γ be a no rmal subgro up. Le t ( X, µ ) b e a s ta ndard Borel space with a probability measure µ , on which Γ acts b y m.p. Borel isomorphisms . F or exa mple, o ne ca n take any measure on { 0 , 1 } and consider the inﬁnite pro duct { 0 , 1 } Γ on whic h Γ a cts b y shifts. Clearly , the translation gr o upo id ( X ⋊ Γ , µ ) is a discrete measured group oid and ( X ⋊ Λ , µ ) is a s trongly normal subgro upo id. If ( X ⋊ Γ , µ ) is erg o dic, then let Q X Λ ⊂ Γ denote the quotient group oid, whic h exists b y Theorem 3.12 . A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 19 5.2. Gab ori au’s extens i on. Let 1 → Λ → Γ p − → Q → 1 b e a shor t exa ct sequence of groups. W e descr ib e a sp ecial case of 5.1 that Gab oriau used to prov e v anishing results for L 2 -Betti n umbers of groups [ 13 ]. Let ( X, µ ) be a Γ-probability space and ( Y , ν ) be a Q -pro bability spac e . Let Z = X × Y be the pro duct of proba bilit y spac es. The gro up Γ acts o n Y via p and o n Z by the dia gonal action. Then ( Z ⋊ Λ , µ × ν ) is a strongly normal subgroup o id o f ( Z ⋊ Γ , µ × ν ). W e refer to (5.1) 1 → ( Z ⋊ Λ , µ × ν ) − → ( Z ⋊ Γ , µ × ν ) − → ( Y ⋊ Q, ν ) → 1 as Gab ori au’s extension ; it is a strong extensio n provided Γ a cts erg o dically on Z . F o r every ergo dic Q -proba bilit y spa c e ( Y , ν ) we can ﬁnd an erg o dic Γ-probability space ( X , µ ) such that Γ acts ergo dically on ( Z, µ × ν ): T ake, for example, the inﬁnite pro duct { 0 , 1 } Γ with the equidistribution on { 0 , 1 } and Γ acting by the shift ( Bernoul li action ). Since this action is mixing, the diagonal Γ- action on Z is still ergo dic. 5.3. The principal extensi on. Let ( G , µ ) b e an er go dic discrete measured gr o up o id. W e denote b y G stab : = { γ ∈ G | r ( γ ) = s ( γ ) } the stabilizer gr oup o id . W e call G x : = { γ ∈ G ; r ( γ ) = s ( γ ) = x } the isotr opy gr oup , o r the stabilizer , of x ∈ G 0 . ( G stab , µ ) is a s trongly norma l subgr o up o id o f ( G , µ ). Indeed, the Borel map ( r × s ) : ( r × s ) − 1 (∆( G 0 )) → G 0 has countable ﬁber s. Hence, w e can use the selection theorem to ﬁnd choice functions . W e denote by ( G rel , µ ) the quotient g roup oid and call it the gr oup oi d of the asso ciate d e quivalenc e r elation . Obviously , ( G rel , µ ) has the sa me unit space G 0 . The strong extension 0 → ( G stab , µ ) → ( G , µ ) → ( G rel , µ ) → 0 is called the princip al ex tension , asso ciated to the discr ete measured group o id ( G , µ ). The discrete mea sured group oid G stab can b e b est v ie wed as a dir e ct int e gr al or Bor el ﬁeld of discrete groups. All functorial constr uc tio ns can b e carried out ﬁb er wise. The following lemma is therefore no s ur prise and w e omit its pro of, which needs a little technical detour . Lemma 5.1. With the notation of the pr evious example, for every p ≥ 0 we have b (2) p ( G stab ) = Z G 0 b (2) p ( G x ) dµ ( x ) . 6. Measura ble cohomol ogical dimensio n W e are now co ming to a more conce ptio nal study of the co homologica l pr op erties of a discrete measured gr oup oid. The main applicatio n of the results in this section is the pro of of the Hopf-Singer C o njecture for poly-sur face groups. 6.1. Deﬁni tion of the measurable cohomolog i cal di m ension. Let ( G , µ ) b e a dis crete measured group oid and Mo d( R ( G , µ )) comp the category of co mplete R ( G , µ )-mo dules. W e deﬁne the me asur able c oho molo gic al dimension mcd C ( G , µ ) to b e the length of the sho r test resolution of M ( G 0 ) by pr o j ective ob jects in Mo d( R ( G , µ )) comp . The fo llowing r esult is classical [ 29 , Lemma 4.1.6 o n p. 9 3]: Theorem 6.1. L et ( G , µ ) b e a discr ete me asur e d gr oup oid. The fol lowing statement s ar e e quiva lent: i) mcd C ( G , µ ) ≤ n . 20 R OMAN S AUER AND ANDREAS THOM ii) F o r al l M ∈ Mo d( R ( G , µ )) comp and m > n , one has Ext m R ( G ,µ )  M ( G 0 ) , M  = 0 . The spectral sequence of Theorem 4.7 immediately yields a pr o duct estimate as follo ws: Corollary 6.2 (of Theorem 4.7 and Remark 4.9 ) . L et 1 → ( S , µ ′ ) → ( G , µ ) → ( Q , ν ) → 1 b e a str ong extension of di scr ete me asur e d gr oup oids or a Gab oria u’s exten s ion ( 5.1 ) . The n, mcd C ( G , µ ) ≤ mcd C ( S , µ ′ ) + mcd C ( Q , ν ) . T ur ning to the in teresting case of tr anslation gr oup oids, we make the following deﬁnition: Deﬁnition 6.3 . Let Γ be a discrete group. W e set: mcd C (Γ) = min { mcd C ( X ⋊ Γ , µ ) | Γ y ( X , µ ) } , where we take the minimum over all m.p. actions of Γ on a standard probability spa c e ( X, µ ). W e call mcd C (Γ) the me asur able c ohomo lo gi c al dimension of Γ. W e need the following lemma. Lemma 6.4. L et Γ b e a discr ete gr oup and let Γ y ( X , µ ) and Γ y ( Y , ν ) me asur e pr eserv- ing actions as ab ove. L et f : Y → X b e a me asur e pr eserving Γ -e quiva riant map. Then, mcd C ( Y ⋊ Γ , ν ) ≤ mcd C ( X ⋊ Γ , µ ) . Pr o of. Clearly , there is a Γ-equiv ariant ring homomorphism f ∗ : M ( X ) → M ( Y ). Let P ∗ → M ( X ) b e a pro jective r esolution in the category Mo d( R ( X ⋊ Γ , µ )) comp . Since M ( X ) is von Neumann regular , M ( Y ) is a ﬂat M ( X )-mo dule. Hence, ( M ( Y ) ⋊ Γ) ⊗ M ( X ) ⋊ Γ P ∗ = M ( Y ) ⊗ M ( X ) P ∗ → M ( Y ) ⊗ M ( X ) M ( X ) = M ( Y ) is a resolution. Setting Q ∗ = ( M ( Y ) ⋊ Γ) ⊗ M ( X ) ⋊ Γ P ∗ and using exactness of comple- tion (with resp ect to the M ( Y )-mo dule structure ), we obtain a resolution Q ′ ∗ → M ( Y ) by co mplete modules . The mo dules Q ′ ∗ are pro jective in Mo d( R ( Y ⋊ Γ , ν )) comp [ 27 , Theo- rem 2.7 (3)].  Prop ositio n 6. 5. L et 1 → Λ → Γ → Q → 1 b e an extensions of gr oups. Then mcd C (Γ) ≤ mcd C (Λ) + mcd C ( Q ) . Pr o of. Let Λ y ( X ′ , µ ′ ) and Q y ( Y , ν ) b e measura ble actions as ab ov e that achiev e the measurable cohomolo gical dimension. Let X b e the coinduction of X ′ , i.e. the Γ- space X = map Λ (Γ , X ′ ), o n which γ ∈ Γ acts fro m the left by co mpo s ition with the Γ- map r γ − 1 : Γ → Γ , γ 0 7→ γ 0 γ − 1 . By choosing a set theor etic section s : Γ / Λ → Γ of the pro jection with s (1) = 1, we obtain a bijection X ∼ = − → Q Γ / Λ X ′ . W e endow X with the structure of a standard pro bability spa c e ( X , µ ) b y pulling back the pro duct measure. This structure do es not dep e nd on the choice o f s ; the measure µ is Γ-inv ariant [ 14 , 3.4]. W e hav e mcd C ( X ⋊ Λ , µ ) ≤ mcd C ( X ′ ⋊ Λ , µ ′ ) b y the previous lemma. Setting Z = X × Y as in Gab oriau’s extensio n (see Subsectio n 5.2 ) yields a translation gr oup oid ( Z ⋊ Γ , µ × ν ) and an extension of discrete mea sured group oids as follo ws: 1 → ( Z ⋊ Λ , µ × ν ) − → ( Z ⋊ Γ , µ × ν ) − → ( Y ⋊ Q, ν ) → 1 A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 21 Again b y the previous lemma, mcd C ( Z ⋊ Λ , µ × ν ) ≤ mcd C ( X ⋊ Λ , µ ). By Corollar y 6.2 w e obtain that mcd C (Γ) ≤ mcd C ( X ′ ⋊ Λ , µ ′ ) + mcd C ( Y ⋊ Q, ν ) . Hence w e completed th e pro of.  The precise relations hip betw een the measurable cohomological dimension to Gab or iau’s ergo dic dimension [ 13 ] is not clear at present. Certainly , the measurable cohomologica l dimension is smaller o r equal to the erg o dic dimension; how ev er, the reverse ineq uality seems to be more diﬃcult to establish. 6.2. The Singer condition . Striving for a conceptional explanation of the Hopf-Singer conjecture in terms of ergo dic theory , w e ma ke the following deﬁnition. Recall that a Poinc ar´ e duality gr oup of dimension n is a group Γ of c ohomologic al dimension n such that H p (Γ , Z Γ) ∼ = ( Z ǫ if p = n , 0 if p 6 = n . Here Z ǫ = Z as an ab elian gro up. The g roup H p (Γ , Z Γ) car ries a natura l right Γ-mo dule structure, and Z ǫ denotes the right Γ-mo dule structur e o n Z that mak es the abov e isomor- phism Γ-equiv arian t. If Z ǫ = Z is the trivial Γ-module, then Γ is called ori entable . Deﬁnition 6.6. Let Γ b e a Poincar´ e duality g roup o f dimension 2 n . W e say that Γ satisﬁes Singer’s c ondition if mcd C (Γ) ≤ n. R emark 6.7 . The deﬁnition could be phrase d in a straig htforward wa y for prop er and co - compact Γ- C W- c o mplexes, sa tis fying equiv a riant Poincar ´ e dua lit y . W e leav e this to the reader. Note that Singer’s condition and ( 4.6 ) imply that all L 2 -Betti num b e r s ab ove the middle dimension v anish. Hence, by Poincar´ e duality , the o nly po ssible non-trivia l L 2 -Betti num ber is in the middle dimension. Note that this is also true in the non-orientable case since every non-o rientable Poincare duality gro up ha s an orientable subgro up of index 2. It is th us of some in terest to under stand the c la ss of g r oups which sa tisfy Singer’s c ondition. F undamental g roups of close d surfaces with genus ≥ 2 a re measure equiv alen t to the free group of r ank 2 [ 14 ]. The free abelian groups Z n are measure eq uiv alent to Z [ 14 ]. Beca us e of Lemma 7.2 we th us obtain the ﬁrst interesting examples of groups satisfying Singer’s condition: Theorem 6.8. L et S g b e the close d orientable surfac e o f genus g ≥ 1 . Then π 1 ( S g ) satisﬁes Singer’s c ondition. Since Poincare dua lit y gr oups are closed under ex tensions [ 4 , Satz 2.5], Prop os itio n 6 .5 yields: Theorem 6.9. L et Λ b e a Poinc ar ´ e duality gr o up of dimension 2 n and Q b e a Poinc ar´ e duality gr oup of dimension 2 m . L et 1 → Λ → Γ → Q → 1 b e an extension of gr oups. Then Γ is a Poinc ar ´ e duality gr oup of dimension 2( n + m ) , and it satisﬁes Singer’s c onditio n if Λ and Q satisfy Singer’s c ondition. 22 R OMAN S AUER AND ANDREAS THOM It is obvious tha t this pr ovides a p ow erful to o l for the co nstruction of gr oups which sa tisfy Singer’s co ndition. Indeed, free pro ducts of amenable groups and surface gr oups (in fact all groups measur e equiv alen t to free g roups) can b e used a s basic building blo cks. A class o f groups which has b een studied to some extend and ﬁts into our fra mework is g iven by the following deﬁnition: Deﬁnition 6.10. A g r oup Γ is said to b e a p oly-surfac e gr oup , if there exists a se r ies of normal subgroups { e } = Γ 0 ⊳ Γ 1 ⊳ · · · ⊳ Γ n = Γ , such that Γ k / Γ k − 1 is a sur face group, i.e. the fundamen tal gro up o f a closed, orientable surface of gen us ≥ 2, for all 1 ≤ k ≤ n . It was proved in [ 17 ] that every p oly- surface gro up a dmits a subgr o up of ﬁnite index, which is the fundamental group of a closed, orient able, aspherical manifold. Hence, the following coro llary is o f impor ta nce. Corollary 6 .11. Poly-surfac e gr oups satisfy S inger’s c ondi tion. In p articular, the Hopf- Singer c onje ctur e holds for close d aspheric a l manifolds with p oly-surfac e fundamental gr oup. 7. P roofs of applica tions 7.1. Pro of of Theorem 1.1 . W e give the pro of o nly in the case where Q = R hyp is an inﬁnite a menable equiv alence re la tion. (The more genera l case o f inﬁnite amena ble measured group oids is more tedious but a na logous.) Ther e exists an increasing chain of ﬁnite sub-relatio ns R n hyp ⊂ R hyp , which are isomorphic to the group oid ( X ⋊ S n ) rel , where S n per mut es a par tition of a contin uous Borel probability space ( X, µ ) into n sets of equal measure. Mor eov er, we ﬁnd these sub-rela tions suc h that R hyp = ∞ [ n =1 R n hyp . In the extension 1 → S → R φ → R hyp → 1 we can take inv erse images R n = φ − 1 ( R n hyp ) and note that n · b (2) p ( R n ) = b (2) p ( S ) . Indeed, this follows from standard homolo gical arguments s inc e R ( R n , µ ) is just an n × n - matrix algebra o ver R ( S , µ ). Since R = ∪ ∞ n =1 R n , w e conclude that b (2) p ( R ) ≤ lim inf n →∞ b (2) p ( R n ) = lim inf n →∞ n − 1 · b (2) p ( S ) . Provided the ﬁrst inequality holds, this would ﬁnish the pro o f since b (2) p ( S ) < ∞ implies now b (2) p ( R ) = 0. The ﬁrst inequalit y follo ws fro m the follo wing three facts: (i) R ( R n , µ ) ⊂ R ( R , µ ) is a dimension ﬂat r ing extension since ∪ n R ( R n , µ ) is ﬂat over R ( R n , µ ) and rank-dense in R ( R , µ ), (ii) L ∞ ( R 0 ) ⊗ R ( R ,µ ) ? = c  colim L ∞ ( R 0 ) ⊗ R ( R n ,µ ) ?  for the same reason that ∪ n R ( R n , µ ) is rank-dense in R ( R , µ ), and A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 23 (iii) dim colim M n ≤ lim inf n →∞ dim M n by [ 21 , Theorem 6.13 (2) on p. 243 ]. By ( i ), ( ii ), and exactness of completio n and directed co -limits, H k ( R , U ( R , µ )) = colim H k ( R n , U ( R n , µ )) = colim  H k ( R n , U ( R n , µ )) ⊗ U ( R n ,µ ) U ( R , µ )  . This implies the claim since, using ( iii ), we get: b (2) k ( R n ) = dim U ( R n ,µ ) H k ( R n , U ( R n , µ )) dim U ( R ,µ ) H k ( R n , U ( R n , µ )) ⊗ U ( R n ,µ ) U ( R , µ ) 7.2. Pro o f of Theorem 1.3 and 1.5 . Consider the strong extension 1 → ( S , µ ) → ( G , µ ) p − → ( Q , ν ) → 1 . The E 1 -term of the corresp onding sp ectr al sequence for is E p,q 1 = hom R ( Q ,ν )  F q ( Q ) , H q ( S ; U ( G , µ ))  , where F ∗ ( Q ) is a pro jectiv e r e solution of M ( Q 0 ) in Mod( R ( Q , ν )) comp . Thu s, using Remark 4.10 , E p,q 1 = 0 for 0 ≤ q ≤ d . It follows tha t E p,q r = 0 for 0 ≤ q ≤ d and 1 ≤ r ≤ ∞ . This implies that E 0 ,d +1 r ∼ = E 0 ,d +1 ∞ for r ≥ 2 and H d +1 ( G , U ( G , µ )) ∼ = E 0 ,d +1 ∞ . Thu s, (7.1) b (2) p ( G ) = 0 for 0 ≤ p ≤ d . and b (2) d +1 ( G ) = dim U ( G , µ ) H 0  Q ; H d +1 ( S ; U ( G , µ )))  (7.2) = dim U ( G , µ ) hom R ( Q ,ν )  M ( Q 0 ) , H d +1 ( S ; U ( G , µ ))  . This co mpletes the pro of o f Theor em 1.3 . F or the proof of Theorem 1.5 , it remains to show that b (2) d +1 ( G ) = 0 provided b (2) d +1 ( S ) < ∞ . Let us a ssume in a ddition that ( Q , ν ) is an inﬁnite equiv alence r elation. Since ( Q , ν ) is inﬁnite, there e x ists an inﬁnite, amenable subr elation R hyp ⊂ Q (see [ 12 , P r o of of Pro p o - sition I I I.3 ; 13 , P r o of of Th´ eor` eme 6.8]). By the same argument as for ( 7.2 ) a pplied to 1 → S → p − 1 ( R hyp ) → R hyp → 1 we get that (7.3) b (2) d +1  p − 1 ( R hyp )  = dim U ( G , µ ) hom R ( R hyp ,ν )  M ( Q 0 ) , H d +1 ( S ; U ( G , µ ))  . Clearly , Equation 7.3 implies b (2) d +1  p − 1 ( R hyp )  ≤ b (2) d +1 ( S ), but this holds also for the inv erse image of a ny ﬁnite index sub-relation R ′ hyp ⊂ R hyp . Hence, since suc h sub-r elations exist for an y giv en index n , n · b (2) d +1 ( p − 1 ( R hyp )) = b (2) d +1 ( p − 1 ( R ′ hyp )) ≤ b (2) d +1 ( S ) for an y n , a nd so b (2) d +1 ( p − 1 ( R hyp )) = 0 as b (2) d +1 ( S ) < ∞ . Since the natura l map hom R ( Q ,ν )  M ( Q 0 ) , H d +1 ( S ; U ( G , µ )  → hom R ( R hyp ,µ )  M ( Q 0 ) , H d +1 ( S ; U ( G , µ )  , is ob viously injectiv e, b (2) d +1 ( G ) = 0 follows now from ( 7.1 ) and ( 7.2 ). 24 R OMAN S AUER AND ANDREAS THOM In the c ourse o f the pro o f, we assumed that ( Q , ν ) was an eq uiv ale nc e rela tion. Again, the more general cas e of inﬁnite discrete measur ed group oids is a bit mor e tedious but ana logous. 7.3. Pro of of Theorem 1.10 . Let us recall the deﬁnition of or bit equiv alence. Deﬁnition 7.1 . W e say that tw o coun table groups Γ and Λ a re orbit e quivalent if there is a proba bilit y spac e ( X, µ ) a nd esse n tially free µ -preserving a ctions of Γ and Λ such that for µ -a.e. x ∈ X we have Γ x = Λ x . F urthermore, Γ and Λ a re we akly orb it e quivalent if there is a pr obability space ( X, µ ), essentially free µ -preserving a ctions of Γ and Λ on X , a nd Bo rel subsets A ⊂ X a nd B ⊂ X such that Γ A = X and Λ B = X up to null sets and suc h tha t for µ -a.e. x ∈ X w e ha ve Γ x ∩ A = Λ x ∩ B . Lemma 7.2. If the gr oups Q 0 and Q 1 ar e me a sur e e quival ent, then mcd C ( Q 0 ) ≤ c d C ( Q 1 ) . Pr o of. F ur man [ 10 ] show ed that groups are measure equiv alen t if and o nly if they a re w eakly orbit equiv alent. Let us ass ume ﬁr st that Q 0 and Q 1 are orbit equiv alen t. Let ( X , µ ) b e a probability spa ce equipp e d with m.p. free a ctions of Q 0 and Q 1 such that the actions have the sa me o rbit equiv a lence re la tion, which coincides with the transla tion gr oup oid. By [ 10 , Lemma 2 .2] one can assume that b oth actions a re ergo dic. So we hav e a n identiﬁcation of the corresp onding group oid rings R ( X ⋊ Q 0 ) ∼ = R ( X ⋊ Q 1 ). Consider the following commutativ e diagram of functors: Mo d( C Q 1 ) / / . . Mo d( L ∞ ( X ) ⋊ Q 1 ) / / Mo d( L ∞ ( X ) ⋊ Q 1 ) comp ∼ =   Mo d( R ( X ⋊ Q 1 )) comp ∼ =   Mo d( R ( X ⋊ Q 0 )) comp The ho r izontal arrows either deno te co mpletion o r ring extension. The vertical arrows are natural is omorphisms of abelia n catego ries. Concerning the uppe r v ertical arr ow, we use that L ∞ ( X ) ⋊ Q 1 is dense in R ( X ⋊ Q 1 ), thus leading to a n identiﬁcation of co mplete L ∞ ( X ) ⋊ Q 1 - with complete R ( X ⋊ Q 1 )-mo dules (see [ 27 , Le mma 4.4]). The ﬁrst hor izontal arrow is an exact functor b ecause o f L ∞ ( X ) ⋊ Q 1 ⊗ C Q 1 M ∼ = L ∞ ( X ) ⊗ C M . The completio n functors ar e exact and preserve pro jectiv es b y [ 27 , Lemma 2.6 and Theorem 2.7]. Star ting with a pr o jective C Q 1 -resolution of C of length n , one obtains by following the a rrows a pro jective resolutio n P ∗ of M ( X ) in Mo d( R ( X ⋊ Q 0 )) comp of length n . The general case where Q 0 and Q 1 are weak orbit equiv alent demands only small mo di- ﬁcations. F or the up c oming discussion of full idempo tent s and Morita equiv a lence we rec- ommend Lam’s b o ok [ 20 , Section 18 ]. Recall that a n idemp otent p in a ring R is called ful l if the elemen ts rpr ′ with r , r ′ ∈ R gener ate R a dditiv ely . Let A ⊂ X and B ⊂ X b e Borel subsets of po sitive measure such that for a.e. x ∈ X we hav e Q 0 x ∩ A = Q 1 x ∩ B . In particular , this gives an identiﬁcation of the restricted translation group oids R ( X ⋊ Q 0 ) | A ∼ = R ( X ⋊ Q 1 ) | B . The characteristic f unctions χ A and χ B are ful l idempotents in the gro upo id rings R ( X ⋊ Q 0 ) and R ( X ⋊ Q 1 ), resp ectively . This is ea sily concluded from ergo dicit y and Lemma 3.8 . Hence A SPECTRAL SE QUENCE FOR GR OUPS AND GR OUPOIDS 25 R ( X ⋊ Q 0 ) and χ A R ( X ⋊ Q 0 ) χ A = R ( X ⋊ Q 0 | A ) are Morita equiv alent, i.e. their mo dule categ o ries a re equiv alent as a b elia n categories. Ex- plicitly , the Mor ita equiv alence is given by M 7→ χ A M . F rom this, it is obvious that the Morita equiv alence restricts to the sub catego ries of complete modules. The same argument holds for Q 1 in plac e of Q 0 . After replacing the low er v ertical arr ow in the diagra m above by the equiv alence o f abelia n categories Mo d( R ( X ⋊ Q 1 )) comp ≃ / /      Mo d( R ( X ⋊ Q 1 ) | B ) comp ≃   Mo d( R ( X ⋊ Q 0 )) comp Mo d( R ( X ⋊ Q 0 ) | A ) comp ≃ o o we can run the same kind of ar gument a s before. This ﬁnishes the pr o of of the lemma.  W e now tur n to the prov e o f Theorem 1.10 in the fo r m needed fo r the application tow ards Theorem 1.15 . Theorem 7.3 . L et 1 → Λ → Γ → Q → 1 b e a short exact se quenc e of gr oups. Supp ose that b (2) p (Λ) = 0 for p > m and let n = mcd C ( Q ) . Then b (2) p (Γ) = 0 for p > n + m . Pr o of. Let Q y ( X , ν ) realize mcd C ( Q ). Consider Gab oriau’s extension (see Subsection 5.2 ) of the form 1 → ( Z ⋊ Λ , µ × ν ) − → ( Z ⋊ Γ , µ × ν ) − → ( X ⋊ Q, ν ) → 1 . The E 1 -term of the co r resp onding spectr a l sequence is (7.4) E p,q 1 = hom R ( X ⋊ Q )  P p , H q ( R ( Z ⋊ Λ); U ( Z ⋊ Γ))  , where P ∗ is a pr o jective resolution of M ( X ) of leng th n in the ca tegory Mo d( R ( X ⋊ Q )) comp (see Remark 4.8 ). Hence E p,q 1 = 0 whenever p > n . Moreover, E p,q 1 = 0 if q > m , s inc e H q ( R ( Z ⋊ Λ); U ( Z ⋊ Γ)) = 0 for q > m , by Remark 4.1 0 and Equa tion ( 4.6 ) . This yields the theorem since E p,q ∞ = 0 for all p + q > n + m , a nd th us b (2) i (Γ) = b (2) i ( Z ⋊ Γ) = 0 for i > n + m .  Clearly , Theorem 7.3 and Lemma 7.2 imply Theore m 1.10 . 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University of Chicago, Chicago, USA Curr ent addr ess : M athematisches Institut der WWU M ¨ unster, Einsteinstr. 62, 48149 M ¨ unste r, Germany E-mail addr ess : sauerr@uni-mue nster.de URL : www.romansa uer.de Ma thema tisc hes I nstitut der Georg-A ugust Un iversit ¨ at G ¨ ottingen, Bunsenstr. 3-5, 37 073 G ¨ ottingen, German y E-mail addr ess : thom@uni-math. gwdg.de URL : www.uni-mat h.gwdg.de/ thom

A spectral sequence to compute L2-Betti numbers of groups and groupoids

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