Splitting formulas for certain Waldhausen Nil-groups

SPLITTING F ORMULAS F OR CER T AIN W ALDHA USEN NIL-GR OUPS. JEAN-FRANC ¸ OIS LAFONT AND IV ONNE J. OR TIZ Abstract. W e provide splitting formulas for certain W aldhausen Nil- groups. W e fo cus on W aldhausen Nil- groups associated to acylindric al amalgamations Γ = G 1 ∗ H G 2 of groups G 1 , G 2 o ver a common subg roup H . F or these amalga- mations, we explain ho w, provided G 1 , G 2 , Γ satisfy the F arr ell-Jones i somor- phism conjecture, the W aldhausen Nil-groups N il W ∗ ( RH ; R [ G 1 − H ] , R [ G 2 − H ]) can be expressed as a direct s um of Nil-groups associated to a specific collection of virtually c yclic subgroups of Γ. A special case co vered b y our theorem is the case of arbitrary amalgamations ov er a finite group H . 1. Introduction W aldhaus en’s Nil- g roups w ere in tro duced in the tw o foundatio nal pap ers [W78a], [W78b]. The motiv ation b ehind these Nil-groups or iginated in a desire to have a May er-Vietoris t yp e sequence in alge br aic K-theory . More precisely , if a gr oup Γ = G 1 ∗ H G 2 splits as an amalgama tion of t wo gro ups G 1 , G 2 ov er a common subgroup H , one ca n a sk how the algebraic K -theory o f the group ring R Γ is related to the algebra ic K-theor y of the in tegra l group rings RG 1 , R G 2 , R H . Motiv ated by the corresp o nding question in homology (or co homology ), one might exp ect a May er-Vietoris t yp e exact sequence: . . . → K i +1 ( R Γ) → K i ( RH ) → K i ( RG 1 ) ⊕ K i ( RG 2 ) → K i ( R Γ) → . . . A ma jor result in [W78 a], [W78b] w as the realization that the Mayer-Vietoris se- quence ab ov e holds, provided one inserts suitable “err or-ter ms”, which are precisely the W aldha usen Nil-group asso ciated to the amalg amation Γ = G 1 ∗ H G 2 . In gen- eral, asso c ia ted to a n y ring S (such as RH ), and any pair of flat R -bimo dules M 1 , M 2 (such as the R [ G i − H ]), W a ldhausen defines Nil-g roups N il W ∗ ( S ; M 1 , M 2 ). The W aldhausen Nil-g r oups N i l W ∗ ( RH ; R [ G 1 − H ] , R [ G 2 − H ]) ar e precisely the “erro r-terms” men tioned above. Another con text in whic h these Nil-gr o ups ma ke an app ear a nce has to do with the r e duct ion t o fi n ites . T o explain this we r ecall t he existence of a generaliz ed equiv ar iant homolog y theor y , having the proper ty that for any gr oup Γ, one has a n isomorphism: H Γ n ( ∗ ; K R - ∞ ) ∼ = K n ( R Γ) . The term a ppe a ring to the right is the homolog y of the Γ-space consisting of a po int ∗ with the trivial Γ-action. N ow fo r any Γ-space X , the ob vious map X → ∗ is clearly Γ-equiv ariant, a nd hence induces a homomo rphism: H Γ n ( X ; K R - ∞ ) → H Γ n ( ∗ ; K R - ∞ ) ∼ = K n ( R Γ) . 1 2 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ The F arrell-Jo nes is omorphism conjecture [FJ9 3] asserts tha t if X = E V C Γ is a mo del for the classifying space for Γ-actions with iso tr opy in the family o f vir- tually cyclic subgroups, then the homomorphis m describ ed ab ov e is a ctually an isomorphism. Explicit mo dels for E V C Γ ar e known for o nly a few class es of groups: virtually cyclic g roups, cr ystallogr aphic groups (Alv ez a nd Ontaneda [A O06] a nd Connolly), hyperb olic gro ups (Juan- Pineda and Leary [JL0 6] and L¨ uck [Lu05]), and relatively hyperb olic gr oups (Lafon t and Ortiz [LO]). In contrast, classifying spaces for pr op er a ctions, denoted E F I N Γ are known for many classes of groups. Now fo r any gro up Γ, one a lwa ys has a unique (up to Γ- equiv ar iant homoto py) ma p E F I N Γ → E V C Γ, which induces a w ell-defined relative assembly map: H Γ n ( E F I N Γ; K R - ∞ ) → H Γ n ( E V C Γ; K R - ∞ ) . In v ie w of the F a rrell-J ones isomorphism conjecture, it is reasona ble to as k whether this latter map is itself a n isomorphism. In many cases, this map is known to b e s plit injective. F o r instance in the case where Γ is a δ -hyperb olic group (see Rosenthal-Sch¨ utz [RS05]), or more genera lly , when Γ has finite asymptotic dimension (see Bartels-Ro senthal [B R]), the relative assembly map is a split injection for arbitra ry rings R . F rom the top ologica l view- po int , the case where R = Z is the most in teresting. In this situation, Bar tels [Ba0 3] has shown that the relativ e as s embly map is split injective for arbitrary groups Γ. If th e relative assembly map discussed abov e is actually an iso morphism, then we say that Γ sa tisfies the r e duction to finites . Let us now sp ecialize to the cas e wher e R = Z , i.e. we will b e fo c us ing on in tegra l group r ings. In this situation, the obstruction to the rela tive assemb ly map b eing an isomor phism lies in the Nil groups asso cia ted to the v ar ious infinite v irtually cyclic subgr oups of Γ . More pr ecisely , the following tw o statements ar e equiv alent (see [FJ93, Thm. A.10]): (1) The gr oup Γ satisfies the reduction to finites. (2) Every infinite virtually cyclic s ubg roup V ≤ Γ satisfies the reduction to finites. Now for a virtually cyclic group V , the failur e of the r eduction to finites ca n b e measured by the cokernel of the relative assembly map. Let us r ecall that infinite virtually cyclic gro ups V come in t wo flavors: • gro ups that surject on to the infinit e dihedr a l group D ∞ , and hence can be decomp osed V = A ∗ C B , wit h A, B , C finite, and C o f index tw o in the groups A, B . • gro ups that do n ot sur ject onto D ∞ , which can a lwa ys b e wr itten in the form V = F ⋊ α Z , where F is a finite group and α ∈ Aut ( F ). In the case where V surjects on to D ∞ , the cok ernel of the r elative a s sembly map coincides with the W aldhaus e n Nil-gro up asso ciated to the splitting V = A ∗ C B . In the case where V do es not s urject o nt o D ∞ , the co kernel of the relative a s sembly map consists of tw o copies of the F a r rell Nil-group asso c ia ted to V = F ⋊ α Z , denoted N K ∗ ( Z F, α ). W e now hav e tw o contexts in which W aldhausen Nil-g roups make an app ear ance: (1) they measure failure of the May er-Vieto ris sequence in alge br aic K -theory , and (2) they contain obstructions for groups to satisfy the r eduction to finites. Having motiv ated our interest in these gr oups, w e can now state our: SPLITTING F ORMULAS FOR CER T AIN W ALDHAUSEN NIL-GR OUPS. 3 Main Theorem. L et Γ = G 1 ∗ H G 2 b e an acylindrica l amalg amation, a nd assume that the F arr el l-Jones Isomorph ism Conje cture holds for the gr oups Γ , G 1 , G 2 . De- note by V a c ol le ction of sub gr oups of Γ c onsisting of one r epr esentative V fr om e ach c onjugacy class o f sub gr oups satisfy ing: (1) V is virtual ly cyclic , (2) V is not c onjugate to a sub gr oup of G 1 or G 2 , and (3) V is maximal with re sp e ct to su b gr oups satisfyi ng (1) and (2). Then for arbitr ary rings R we have a n isomorph ism for ∗ ≤ 1 : N il W ∗ ( RH ; R [ G 1 − H ] , R [ G 2 − H ]) ∼ = M V ∈V H V ∗ ( E F I N V → ∗ ; K R - ∞ ) wher e H V ∗ ( E F I N V → ∗ ; K R - ∞ ) ar e the c okernels of the r elative assembly maps asso ciate d to t he virtu al ly cycli c sub gr oups V ∈ V (and henc e c onsist of classic al F arr el l or Waldhausen Nils). Remarks: (1) The notion of a n acyclindric al amalgamation w a s formulated b y Sela [Sel97] in rela tion to his work o n the access ibilit y problem for finitely generated groups. An amalgamation Γ = G 1 ∗ H G 2 is said to be acylindrical if there exists a n int eger k such that, fo r every path η of length k in the Bass-Ser re tree T as so ciated to the splitting of Γ, the stabilizer of η is finite. Observe that if the amalg a mating subgroup H is finite, the amalgama tio n is automatically acylindrical (with k = 1). (2) The cokernels H V i ∗ ( E F I N V i → ∗ ; K R - ∞ ) are the familiar W aldhause n or (t w o copies of the) F arrell Nil groups, according to whether the virtually cyclic g roup V i surjects onto D ∞ or not. Note that every virtually cyclic subgro up V that maps onto D ∞ contains a canonical index tw o subgr o up V ′ which do es not map o nt o D ∞ (the pre- ima ge of the obvious index t w o Z subgroup in D ∞ ). Recent indep endent work b y v arious author s (Davis [D], Davis-Khan-Ranicki [DKR], Quinn, and Reich), has established that the W aldhausen Nil-g roup of V is isomo rphic to the F arr e ll Nil-group of V ′ . (3) F rom the computational viewp oint, the Main Theorem com bined with the pre- vious remar k completely reduces (mo dulo the Isomorphis m Co njectur e) the co mpu- tation of W aldhausen Nil-groups asso ciated to acylindrical amalg amations to tha t of F arr ell Nil-groups. (4) Consider the simple case of a fr e e pr o duct Γ = G 1 ∗ G 2 . In this s ituation, the group Γ is known to be (strongly) relatively h yp erb olic, rela tive to the subgroups G 1 , G 2 . Assuming the F a rrell-Jo nes isomorphism conjecture for Γ, previo us w ork of the authors [LO2, Cor. 3.3] yields the f ollowing expression for K n ( R Γ): H Γ n ( E F I N Γ) ⊕  M i =1 , 2 H G i n ( E F I N G i → E V C G i )  ⊕  M V ∈V H V n ( E F I N V → ∗ )  where V is the co llection o f virtua lly cy c lic subgro ups mentioned in our Main Theo- rem (w e omitted the coefficients K R - ∞ to simplify notation). No w, morally speak- ing, the W aldhausen Nil-group is the p ortion of the K - theory of Γ tha t do es not c ome fr om the K -the ory of t he factors G i . Recalling the well known fact that every finite subgr oup of Γ has to b e co njugate into one o f the G i , one sees that in the expression a bove, this s ho uld just be the last term. Our Main Theorem ca me a bo ut from trying to make this heuristic precise. 4 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ 2. Proof of Main Theorem Given the group Γ = G 1 ∗ H G 2 satisfying the hypo theses o f o ur theorem, let us form the f amily F of subgr oups of Γ co nsisting of all virtually cyclic subgr oups that can b e conjugated into either G 1 ≤ Γ or G 2 ≤ Γ. Observe that we have a containmen t of families F ⊂ V C of subgroups o f Γ, which in turn induce an assembly map: ρ : H Γ n ( E F Γ; K R - ∞ ) → H Γ n ( E V C Γ; K R - ∞ ) . Our proof will focus on analyzing the map ρ , and in par ticular on ga ining a n un- derstanding of the cokernel of that map. Let us start b y describing the W aldhausen Nil-group as the cokernel of a suitable a ssembly map. Cor resp onding to the splitting Γ = G 1 ∗ H G 2 , we hav e a simplicial ac tio n of Γ on the corresp onding Bas s -Serre tree T (s e e [Ser80]). F ro m the natural Γ-equiv ariant map T → ∗ , w e get an asse mbly map: ρ ′ : H Γ n ( T ; K R - ∞ ) → H Γ n ( ∗ ; K R - ∞ ) ∼ = K ∗ ( R Γ) . W e s ta rt out with the imp ortant: F act: The map ρ ′ is split injective, and coker( ρ ′ ) ∼ = N il W ∗ ( RH ; R [ G 1 − H ] , R [ G 2 − H ]) . A proof of this F act can b e found in Davis [D, Lemma 7] (see also Remark (i) at the end o f this section). In view of this result, w e ar e mer ely trying to identify the cokernel of the map ρ ′ . The first s tep is to r elate the cokernel of ρ ′ with the cokernel o f the map ρ . Claim 1: The map ρ is split injective, and there is a canonica l isomorphism coker( ρ ) ∼ = coker ( ρ ′ ) W e obser ve that we hav e four families o f subgro ups that we are dealing with: the three we hav e lo oked at so far a r e V C , ALL , and the family F we intro duce d a t the beg inning of our pro of (cons isting of vir tually cyclic subgroups conjugate in to one of the G i ). In addition, there is the family G consisting of al l subgroups of Γ which can be conjugated in to either G 1 or G 2 . No w observe that we hav e con tainment s of families F ⊂ G ⊂ ALL , a nd F ⊂ V C ⊂ ALL . F urther more, we ha ve that T is a mo del for E G Γ. This yields the following comm utative diag ram: H Γ ∗ ( E F Γ; K R - ∞ ) ρ / /   H Γ ∗ ( E V C Γ; K R - ∞ ) ∼ =   H Γ ∗ ( T ; K R - ∞ ) ρ ′ / / H Γ ∗ ( ∗ ; K R - ∞ ) where all the maps are relative a ssembly maps corresp onding to the inclusions of the v arious families of subgroups. Note that th e horizont al maps are precisely the ones w e ar e try ing to relate. Now recall that w e are ass uming that Γ satisfies the F ar rell-Jones isomor phism conjecture. This immediately implies that the s econd vertical map is an isomorphism, as indicated in the co mm utative diagram. So in order to iden tify the cok ernels of the t wo horizontal maps , w e ar e left with showing that the first vertical map is also an isomorphis m. SPLITTING F ORMULAS FOR CER T AIN W ALDHAUSEN NIL-GR OUPS. 5 The firs t vertical map is a rela tive assembly map, corresp onding to the inclusion of the families F ⊂ G of subg roups of Γ. In order to show that the relative assembly map is an isomorphism, o ne merely needs to establish that for every maximal subgroup H ∈ G − F , the corresp onding relativ e assembly ma p induced b y the inclusions of families F ( H ) ⊂ G ( H ) of subgro ups of H is a n isomo r phism (see [FJ93, Thm. A.1 0 ], [LS, Thm. 2 .3]). But observe that the maximal subgroups in G − F are precisely the ( conjugates of ) t he subgroups G i ≤ Γ. F ur thermore, for these subgroups, we hav e th at G ( G i ) = ALL ( G i ), and tha t F ( G i ) = V C ( G i ). Hence the relative assembly maps which we req uire to be isomorphisms are exa ctly those induced by E V C G i → E ALL G i ∼ = ∗ , i.e . those that ar ise in the F a rrell- Jones isomor phism conjecture. Since we are assuming the isomorphism conjecture holds for the groups G 1 , G 2 , we conclude that the first vertical map is indeed an isomorphism, completing the pro of of the claim. A t this p oint, combining Claim 1 with the F act , we ha ve a n iden tification: coker ( ρ ) ∼ = coker( ρ ′ ) ∼ = N il W ∗ ( RH ; R [ G 1 − H ] , R [ G 2 − H ]) . In order to complete the pr o of, w e now fo cus entirely on studying the ma p ρ , with a goal of showing that one can express its co kernel as a direct sum of the desired Nil- groups associa ted to the virtually cyclic subgr oups V ∈ V . W e remind the rea der that ρ is the relativ e assem bly ma p induced by the map E F Γ → E V C Γ, where F is the family of subgroups c o nsisting of all virtually cyclic subgroups of Γ tha t can be conjugated in to either G i . In order to analy ze this relative as s embly ma p, we will nee d to make use of s ome prop erties o f the Γ-action o n the Bass-Serre tre e . P articular ly , w e would like to understand the b ehavior of virtually cyclic subgroups V ∈ V C − F . The sp ecific result we will require is con tained in our : Claim 2: In the case of an acylindric al amalgamation, the stabilizer of a ny geo des ic γ in the Bas s-Serre tree T is a virtua lly cyclic subgroup o f Γ. F urthermore, ev ery virtually cyc lic subgroup V ≤ Γ satisfying V ∈ V C − F stabilizes a unique geo desic γ ⊂ T . Let us start by rec alling some basic facts concer ning the a ction of Γ on the Bass-Ser re tr ee T co rresp onding to the amalga mation Γ = G 1 ∗ H G 2 : • the action is witho ut in versions, i.e. if an element stabilizes an edge e , then it automatically pre serves the chosen orientation o f e , • the stabilizer of an y vertex v ∈ T is is omorphic to a conjugate o f G 1 or G 2 , • the stabilize r of any edge e ⊂ T is isomorphic to a conjugate of H , • any finite subgroup of Γ fixes a vertex in T , The first three statements a b ov e are built in to the definition of the Ba ss-Serre tree (see [Ser80]), while the last statement is a well known general facts ab out group actions on trees . W e re mind the reader that a ge o desic in a tree T will be a s ubco mplex s implicially isomorphic to R , with the standard s implicial structure (i.e. v ertices at the integers, and edges b etw een). T o show the first statemen t in our c laim, w e note that S ta b Γ ( γ ) clearly fits in to a short exact sequence: 0 → F i x Γ ( γ ) → S t a b Γ ( γ ) → S i m Γ ,γ ( R ) → 0 6 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ where F ix Γ ( γ ) is the subg r oup fixing γ p oint wise, and S im Γ ,γ ( R ) is the induced simplicial action on R (o bta ined by simplicially iden tifying γ with R ). Note that the gr oup of simplicia l a utomorphisms of R is D ∞ , the infinite dihedral group. In particular, we see that S im Γ ,γ ( R ) is virtually cyclic (in fact is isomorphic to Z / 2 , Z , or D ∞ ). Next w e obser ve that F i x Γ ( γ ) is finite. T o see this, we recall that the amal- gamation Γ = G 1 ∗ H G 2 was as sumed to b e acylindric al , which means that there exists an in teger k ≥ 1 with the property that the stabilize r S tab Γ ( η ) ≤ Γ of a ny combinatorial path η ⊂ T of length ≥ k is finite. Since γ ⊂ T is a ge o desic, it contains combinatorial subpaths η of arbitrarily long length (in particular, length ≥ k ). The ob vious containemen t F i x Γ ( γ ) ≤ S tab Γ ( η ) no w completes the ar gument for the first statement in our claim. F or the second statemen t, we note that Γ acts simplicia lly on t he Bass-Ser re tree, and hence the given virtually cy c lic subgroup V ∈ V C − F likewise inher its an action on T . Since V / ∈ F , we hav e that the V actio n on T has no globally fixed po int . In particular, V must b e an infinite virtually cyclic subgroup, and hence contains elements of infinite order. If g ∈ V is an arbitrary elemen t of infinite order, we now claim that g cannot fix any v ertex in T . Assume, by wa y of c o ntradiction, that ther e ex is ts a vertex v fixed by g (and hence b y V ′ ). Let T ′ ⊂ T b e the subset consisting of po in ts that are fixed b y V ′ . Note that T ′ is non-empt y (s inc e v ∈ T ′ ), and is a subtree of T (since Γ acts simplicially on T ). F urthermore, obser ve that the gro up F := V /V ′ inherits a simplicial action on T ′ . But no te that F is finite, and hence the F -action on T ′ has a fixe d v ertex w ∈ T ′ ⊂ T . But this immediately implies that the orig ina l g r oup V fixes w , a con tradiction as V / ∈ F . Now establishing that V stabilizes a g e o desic is a straig htf orward applica tion of standard techniques in the g eometry of gr oup actions o n trees (applied to T ). F o r the conv enience of the rea der, we give a quick o utline of the arg umen t. F or an ar bi- trary element g o f infinite order in V , one can loo k at the asso ciated displac e ment function on T , i.e. the distance from v to g · v . The previo us par agra ph es tablishes that this function is strictly positive. One then considers the set M in ( g ) of p oints in T which minimize the dis pla cement function, call this minimal v a lue µ g . It is easy to see that: • M i n ( g ) contains a geo des ic γ : tak e a v ertex v ∈ M in ( g ), and consider γ := S i ∈ Z g i · η , wher e η is the geo desic seg ment from v to g · v (note that such a non-trivial segment exists by the previous para graph), • in fact, γ = M in ( g ): a ny po int at distance r > 0 from γ w ill be displaced 2 r + µ g > µ g , and so cannot lie in M in ( g ), • for any non-zero in teger i , M in ( g ) = M in ( g i ): an y point a t distance r > 0 from γ will b e displaced 2 r + | i | · µ g > | i | · µ g , while p oints o n γ will clea rly only b e displa ced | i | · µ g by the e le men t g i , • for an y tw o element s g , h of infinite order in V , we have M in ( g ) = M in ( h ): t wo such elements ha ve a commo n power, and a pply the previo us statement. F ro m the o bserv ations ab ove, we see that every single element in V of infinite order stabilizes the exact sa me g e o desic γ ⊂ T . Hence the only elements w e might ha ve to worry about a re element s h ∈ V of finite or der . F or thes e , we just note that V con tains V ′ ⊳ V , a finite index cyclic normal subgr oup g enerated b y a n elemen t g of infinite o r der. W e have a SPLITTING F ORMULAS FOR CER T AIN W ALDHAUSEN NIL-GR OUPS. 7 natural mor phism fro m H = h h i to Aut ( V ′ ) ∼ = Z / 2. In particular, we hav e that hg h − 1 = g ± 1 and hence we hav e the obvious equalities d ( g · hv , hv ) = d ( h − 1 g h · v , v ) = d ( g ± 1 · v , v ) = µ g ± 1 = µ g . Since hv is minimally displa ced by g , it must lie on γ = M in ( g ). This deals with elements of finite order, hence completes the verification that γ is V -inv ariant. Finally , from the fact th at V has a finite index subgroup that acts on γ via a translation, it is easy to see that ther e are no other V -in v a riant g eo desics in T , yielding uniqueness. This finishes the arg ument for o ur Claim 2 . Having established some ba sic prop er ties of the Γ-a ction on T , we now return to the main a rgument. Rec a ll that by combining our Claim 1 with the F act , w e hav e reduced the pro of of the Ma in Theorem to understanding the cokernel of the relative a ssembly map: ρ : H Γ n ( E F Γ; K R - ∞ ) → H Γ n ( E V C Γ; K R - ∞ ) where F is the family of subg roups consisting of all virtua lly cyclic subgroups of Γ that can b e conjuga ted into either G i . Now r e call that in [LO], the authors introduced the notion of a collection of subgroups to b e adapte d to a nested family of subg r oups, and in the pres ence o f an adapted family , show ed ho w a model for the classifying space with isotropy in the smaller family could be “pr o moted” to a model for the classifying space with isotropy in the lar ger family . I n a subsequent pap er [LO 2], the authors used so me recent work of Lueck-W eiermann [L W] to give an alternate mo del for this classifying space, whic h ha d the additiona l adv a nt age of pr oviding explicit sp littings for the cokernel of the rela tive assembly maps. Let us briefly recall the relev a nt definitions. Given a nested pair of families F ⊂ ˜ F of subgroups of Γ, we say that a collectio n { H α } α ∈ I of subgroups of Γ is adapte d to the pair ( F , ˜ F ) provided that: (1) F o r all H 1 , H 2 ∈ { H α } α ∈ I , either H 1 = H 2 , or H 1 ∩ H 2 ∈ F . (2) The co llection { H α } α ∈ I is c onjugacy clo se d i.e. if H ∈ { H α } α ∈ I then g H g − 1 ∈ { H α } α ∈ I for all g ∈ Γ. (3) Every H ∈ { H α } α ∈ I is self-normalizing , i.e. N Γ ( H ) = H . (4) F o r all G ∈ ˜ F \ F , there e xists H ∈ { H α } α ∈ I such that G ≤ H . In [LO2], we applied this result to the nested family F I N ⊂ V C for relatively hyperb olic gro ups (for which an adapted collection of subgr o ups is easy to find). In our present con text, w e would like to find a collectio n of subgroups adapted to the nested pair of families F ⊂ V C . Recall that F consists of a ll virtually cyclic subgroups that ca n b e conjugated into either G i , and V C consists of all virtually cyclic subgroups of Γ. Claim 3: The collection { H α } of subgroups o f Γ co nsisting o f all maximal virtually cyclic subgroups in V C − F is adapted to the pair ( F , V C ). T o verify this, we first note that pro p e r ties (2 ) and (4) in the definition of an adapted family are immediate. Prop erty (3) follows easily from Clai m 2 : let V ∈ { H α } b e g iven, and consider N Γ ( V ). W e know that V lea ves in v aria n t a unique geo des ic γ ⊂ T . F urthermor e, for every g ∈ Γ, we see that g V g − 1 leav es g · γ inv ar iant. Uniqueness of the V -in v ariant g eo desic γ now implies that γ is ac tually N Γ ( V )-inv a riant. In particular , w e have containmen ts V ≤ N Γ ( V ) ≤ S tab Γ ( γ ). 8 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ But from Claim 2 , we kno w that S tab Γ ( γ ) is virtually cyclic, and maximalit y of V now force s all the containmen ts to be equalities, and in particular, V = N Γ ( V ) as required by pr op erty (3). F or prop erty (1), let V 1 , V 2 ∈ { H α } . W e wan t t o establish that either V 1 = V 2 , or that V 1 ∩ V 2 ∈ F . So let us a ssume that V 1 6 = V 2 . W e know from Claim 2 that each V i stabilizes a unique geo desic γ i , and from the ma ximality o f the groups V i , we actually have V i = S tab Γ ( γ i ). Since V 1 6 = V 2 , w e have that γ 1 6 = γ 2 . Ther e are now tw o possibilities: (i) either γ 1 ∩ γ 2 = ∅ , or (ii) γ 1 ∩ γ 2 is a path in T . W e claim that in b oth cases , the intersection H = V 1 ∩ V 2 ≤ V i has the pro per ty that the H -action o n the corres po nding γ i fixes a p oint . T o see this, let us first consider pos sibility (i): since γ 1 ∩ γ 2 = ∅ , one can consider the (unique) minimal length g eo desic segmen t η joining γ 1 to γ 2 . W e observe that, since H stabilizes bo th γ i , it must leave the segmen t η inv ariant. In particular, H m ust fix the vertex v i = η ∩ γ i ∈ γ i , as desired. Next consider p ossibility (ii): if γ 1 ∩ γ 2 6 = ∅ , then the in tersection will be a subpath η ⊂ γ i . Note that η is either a geo desic se gment , or is a geo desic r ay , and in b oth cases, will be inv ariant under the gr oup H . It η is a geodesic ra y (i.e. ho meo morphic to [0 , ∞ )), then there is a (topo logically) distinguished p oint inside η , whic h will hav e to b e fixe d b y H . If η is a g e o desic segment, then ea ch element in H either fixes η , or reverses η (note that the latter can o nly o cc ur if η has even length, as Γ acts on T without in versions). In particular , w e see that if η has o dd length, then every p oint in η is fixed b y H , while if η has ev en leng th, then the (co mbin ator ial) midpoint is fixed. Finally , we observe that H ≤ V i acts on γ i , and fixes a point. This immediately implies that H contains a subgroup of index at mos t tw o which acts trivial ly on γ i , i.e. H ′ ≤ F ix Γ ( γ i ). But r ecall that the latter gr oup is finite (see the pro o f o f Claim 2 ), co mpleting the pr o of of pr op erty (1). W e c onclude that the collection { H α } is an adapted collection of subg roups for the nested families ( F , V C ), as desired. Finally , w e exploit the adapted family we just constructed to establish: Claim 4: W e have an identification: coker ( ρ ) ∼ = M V ∈V H V ∗ ( E F I N V → ∗ ; K R - ∞ ) where the g r oups H V ∗ ( E F I N V → ∗ ; K R - ∞ ) are the cokernels of the relative a ssem- bly maps asso ciated with the virtually cyclic gro ups V ∈ V The a r gument for this is vir tually identical to the one given in [LO2, Co r. 3 .2 ]; we repro duce the a rgument here for the conv enience of the rea der. F r om Claim 3 , w e ha ve an adapted f amily for t he pair ( F , V C ), consisting of all maximal sub- groups in V C − F . F ro m this ada pted collectio n of subgroups, [LO2, Prop. 3.1] establishes (using [L W, Thm. 2.3 ]) a method for constructing an E V C Γ. Namely , if V is a complete set of representativ es of the co njugacy classes within the a dapted collection of subgroups { H α } , form the cellula r Γ-pushout: ` V ∈V Γ × V E F ( V ) α   β / / E F (Γ)   ` V ∈V Γ × V E V C ( V ) / / X SPLITTING F ORMULAS FOR CER T AIN W ALDHAUSEN NIL-GR OUPS. 9 Then the resulting space X is a mo del for E V C (Γ) (w e refer the r eader to [LO 2, Prop. 3 .1] for a mo re precise discus sion of this result, including a description of the maps α, β in the ab ove cellula r Γ-pushout). Note that the map ρ whose co kernel we are trying to under stand is precisely the map on (equiv ariant) homology induced by the s econd vertical a rrow in the above cellular Γ-pushout. Since X is t he double mapping cylinder of the maps α, β in the abov e diagram, one has a natural Γ- equiv a r iant decompos ition of X b y taking A (resp ectively B ) to b e the [0 , 2 / 3) (resp ectively (1 / 3 , 1]) p ortions o f the double mapping cylinder. Applying the homolo gy functor H Γ ∗ ( − ; K R - ∞ ) (and omitting the co efficients to shorten notation), we ha ve the Ma yer-Vietoris sequence: . . . → H Γ ∗ ( A ∩ B ) → H Γ ∗ ( A ) ⊕ H Γ ∗ ( B ) → H Γ ∗ ( X ) → H Γ ∗− 1 ( A ∩ B ) → . . . But now obser ve that w e have obvious Γ-equiv ariant homo topy eq uiv alences be- t ween: (1) A ≃ Γ ` V ∈V Γ × V E V C ( V ) (2) B ≃ Γ E F (Γ) (3) A ∩ B ≃ Γ ` V ∈V Γ × V E F ( V ) F urther mo re, the ho mo logy theory w e have takes disjoint unions into direct sums. Combining this with the induction structure, we obtain isomorphisms: H Γ ∗ ( A ) ∼ = M V ∈V H Γ ∗ (Γ × V E V C ( V )) ∼ = M V ∈V H V ∗ ( E V C ( V )) H Γ ∗ ( A ∩ B ) ∼ = M V ∈V H Γ ∗ (Γ × V E F ( V )) ∼ = M V ∈V H V ∗ ( E F ( V )) Now observing that the g roups V ∈ V are all virtua lly cyclic, we hav e that each E V C ( V ) can b e taken to be a point. F urther more, for the gr o ups V ∈ V , w e ha v e that the restriction of the family F to V co incides with the family of finite subgroups of V , i.e. E F ( V ) = E F I N ( V ). Substituting all of this in the ab ov e Ma yer-Vietoris sequence, we get the long exact sequence: . . . → M V ∈V H V ∗ ( E F I N ( V )) → H Γ ∗ ( E F Γ) ⊕ M V ∈V H V ∗ ( ∗ ) → H Γ ∗ ( E V C Γ) → . . . Now observe tha t the each of the ma ps H V ∗ ( E F I N ( V )) → H V ∗ ( ∗ ) are s plit injective (see for ins tance [RS05]). Since the map ρ : H Γ ∗ ( E F Γ) → H Γ ∗ ( E V C Γ) is also split injectiv e (from Cl aim 1 ), w e now hav e a n iden tification: coker ( ρ ) ∼ = M V ∈V coker  H V ∗ ( E F I N ( V )) → H V ∗ ( ∗ )  Finally , combining the F act with Claim 1 and Claim 4 , w e see that we have the desired splitting: N il W ∗ ( RH ; R [ G 1 − H ] , R [ G 2 − H ]) ∼ = M V ∈V H V ∗ ( E F I N V → ∗ ; K R - ∞ ) where the groups H V ∗ ( E F I N ( V ) → ∗ ; K R - ∞ ) denote the cokernels app earing in Claim 4 . This completes the pro of of the Main T he o rem. Remark: (1) O ne of the k ey ingredient s in our proo f w as the F act established by Davis in [D, Lemma 7]. Pr ior to learning of Davis’ preprint, the authors had a n 10 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ alternate arg umen t for the F act . F or t he sak e of the interested reader s, w e briefly outline our alternate appro ach. Anderson-Munkholm [AM00, Section 7] defined a functor , c o ntin uo usly con- trolled K -theor y , from the category of diagrams o f holink t y p e to the categor y o f sp ectra. Munkho lm-Prass idis [MP01, Theo rem 2 .1] show ed that the W a ldhausen Nil-group w e are in terested in can b e iden tified with the cok ernel of a natural split injectiv e map ˜ K cc ∗ +1 ( ξ + ) → K ∗ ( Z Γ), where ξ + is a suitably defined diagr am of holink type asso cia ted to the splitting Γ = G 1 ∗ H G 2 (see [AM00 , Section 9]). F urther mo re, Anderson-Munkholm hav e sho wn [AM00, Theorem 9.1] that there is a natura l isomorphis m ˜ K cc ∗ +1 ( ξ + ) ∼ = ˜ K bc ∗ +1 ( ξ + ), where the latter is the b oundedly controlled K -theory defined by Ander s on-Munkholm in [AM90]. Finally , there ar e A tiyah-Hirzebruch sp ectra l sequences co mputing both the g r oups ˜ K bc ∗ +1 ( ξ + ) (see [AM90, Theorem 4.1]) and H Γ ∗ ( T ; KZ - ∞ ) (see [Qu82, Section 8]). It is ea sy to v er- ify that the tw o s pec tr al sequences are c anonic al ly identic al : they hav e the s ame E 2 -terms and the same differentials. Combining these r esults, and keeping tr a ck of the v arious maps app ear ing in t he sequence of is omorphisms, o ne can give an alternate pro of of the F act . (2) W e also p oint out that, f rom the Γ- action on the Bass-Serre tree T , it is ea sy to obtain co ns traints on the isomorphism t yp e of g roups inside the collection V . Indeed, any such gr oup m us t b e the stabilizer of a bi-infinite geo desic γ ⊂ T (s e e Claim 2 ), and m ust a ct cocompactly on γ . Reca ll that infinite virtually cy clic subgroups are of tw o types: tho s e that surject o nt o the infinite dihedral gr oup D ∞ , and those who don’t. The groups that surject onto D ∞ alwa ys split as an amalgamation A ∗ C B , with all three groups A, B , C finite, and C of index tw o in both A, B . Obser ve that if V ∈ V is of this type , then under t he action of V on γ ⊂ T , A and B can b e ident ified with the sta bilizers o f a pair of vertices v , w , and C ca n be iden tified w ith the stabilizer of the s egment joining v to w . In particula r, C must b e a su b gr oup of an e dge stabilizer , and hence is conjugate (within Γ) to a finite subgroup of H . F urther mo re, since A, B both stabilize a pair of vertices, they must be co njugate (within Γ) to a finite subg roup of either G 1 or G 2 . The groups that do not surject o nt o D ∞ are of the form F ⋊ α Z , where F is a finite group, and α ∈ Aut ( F ) is an a utomorphism. If V ∈ V is of this form, then for th e action o f V on γ ⊂ T , one has that F c a n b e ident ified with the subset of V that p oint wise fixes γ , while the Z comp onent acts o n γ via a transla tion. In particular, F is again conjugate (within Γ) to a finite subgroup of H . In particular , if w e are given an explicit amalgamation Γ = G 1 ∗ H G 2 , and we hav e knowledge of the finite subgr oups inside the gro ups H , G 1 , G 2 , then we can readily identify up to isomorphism the pos sible g roups arising in the co llection V . If one has knowledge of the Nil-gro ups asso cia ted with these v ar ious gr oups, o ur Main Theorem can be use d to get corresp onding information ab out the W aldhausen Nil-group asso ciated to Γ. 3. Concluding Remarks Having completed the proo f of our Main Theorem, we no w pro ceed to isolate a few in teresting corolla ries. As mentioned ear lier, from the viewpoint of to p o lo gical applications, the most interesting situation is the case where R = Z , i.e. integral group rings. SPLITTING F ORMULAS FOR CER T AIN W ALDHAUSEN NIL-GR OUPS. 11 Corrollary 1. L et Γ = G 1 ∗ H G 2 b e an ama lgamation, and assume that t he F arr el l- Jones Isomorphism Conje ct u r e hol ds for the gr oups Γ , G 1 , G 2 , and that H is finit e. Then the asso ciate d Waldhausen N il-gr oup N il W ∗ ( Z H ; Z [ G 1 − H ] , Z [ G 2 − H ]) is either t rivial, or an infinitely gener ate d t orsion gr oup. Pr o of. Note that since H is finite, the amalgamatio n is acy lindr ical, and o ur Main Theorem applies. So the W a ldhausen group w e are in terested in splits as a direct sum of Nil-g roups assoc iated to a particula r collection V of virtually cyclic sub- groups. It is well known that the Nil-groups asso ciated to virtually cyclic gr oups are either trivial or infinitely generated ( see [F77], [G1], [R]). F urthermore, these groups are known to be purely torsion (see [W e 81], [CP0 2], [KT0 3], [G2]), giv ing us the second statement. A sp ecial case of the abov e corollary is worth mentioning: Corrollary 2. L et Γ = G 1 ∗ H G 2 b e an amalgamation, a nd assume that G 1 , G 2 , and H ar e al l finite. Then the asso ciate d Waldha usen Nil-gr oup N il W ∗ ( Z H ; Z [ G 1 − H ] , Z [ G 2 − H ]) is either trivial, or a n infinitely g ener ate d torsion g r oup. Pr o of. G 1 , G 2 trivially satisfy the F arrell-J ones isomo rphism conjecture, as they are finite. F urthermor e, the gr oup Γ is δ -hyper bo lic, and hence b y a rece n t result of Bartels -L ¨ uc k-Reich [BLR], also satisfies th e isomorphism conjecture. Hence the hypotheses of our previo us corollary are satisfied. Finally , we o bserve that the Bar tels -L ¨ uc k-Reich result [BLR] es ta blishes the Isomorphism conjecture in all dimensions i and arbitr ary rings with unity R . In particular, the hypotheses of our Ma in Theorem hold for arbitrar y amalgamations of finite groups, giving : Corrollary 3. L et Γ = G 1 ∗ H G 2 b e an amalgamation, wher e G 1 , G 2 , and H ar e al l finite. Then, for arbitr ary rings with unity R , we have i somorphisms: N il W ∗ ( RH ; R [ G 1 − H ] , R [ G 2 − H ]) ∼ = M V ∈V H V ∗ ( E F I N V → ∗ ; K R - ∞ ) wher e H V ∗ ( E F I N V → ∗ ; K R - ∞ ) ar e the c okernels of the r elative assembly maps asso ciate d t o the vi rtual ly cycli c sub gr oups V ∈ V , and the c ol le ction V is as in the statement of our Mai n The or em. W e p oint out that the sp ecial case of the mo dular gro up Γ = P S L 2 ( Z ) = Z 2 ∗ Z 3 has also been independently studied b y Da vis-Khan-Ra nicki (see [DKR, Section 3.3]). References [AO06] A. A lve s, P . 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