aa r X i v : . [ m a t h . G R ] F e b RELATIVE ENDS, ℓ -INVARIANTS AND PROPERTY (T) ADITI KAR AND GRAHAM A. NIBLO
Abstract.
We establish a splitting theorem for one-ended groups H ≤ G such that ˜ e ( G, H ) ≥ H is aproper subgroup of G . This yields splitting theorems for groups G withnon-trivial first ℓ -Betti number β ( G ). We verify the Kropholler Con-jecture for pairs H ≤ G satisfying β ( G ) > β ( H ). We also prove thatevery n -dimensional Poincar´e duality ( P D n ) group containing a P D n − group H with property (T) splits over a subgroup commensurable with H . In this article we explore the relationship between the theory of relativeends, groups with non-trivial first ℓ -cohomology and the presence of sub-groups with property (T). The desired conclusion is to obtain splittings ofgroups, i.e., nontrivial decompositions of groups into amalgams or HNN ex-tensions. We use two different notions of ‘relative ends’ for groups H ≤ G ,the geometric one which is usually written e ( G, H ) and its algebraic coun-terpart ˜ e ( G, H ).The classical theory of the ends of a group originated in the work ofFreudenthal and Hopf (See [3], [4]). From the point of view of a geometricgroup theorist the number of ends of a finitely generated group G , written e ( G ), is the number of Freudenthal-Hopf ends of a connected locally finiteCayley graph for G , regarded as a 1-dimensional simplicial complex. While a priori the number could depend on the generating set chosen, it is in factindependent provided the chosen generating set is finite, i.e., it is a quasi-isometry invariant of the group. There is an alternative definition of e ( G )which is more obviously independent of choice of generating sets, and whichextends to a definition of the number of ends for an arbitrary discrete group. Definition 1.
Let G be a discrete group, P ( G ) denote the power set of G ,and F ( G ) denote the set of finite subsets of G . Then F ( G ) , P ( G ) and thequotient F ( G ) \P ( G ) are all F G -modules, where F denotes the field of elements. We denote by e ( G ) the dimension of the G invariant subspace ( F ( G )) \P ( G )) G . Hopf showed in [4] that the number of ends of a finitely generated groupmust be 0, 1, 2 or ∞ . Moreover, groups with 0 and 2 ends are easily classified: e ( G ) = 0 if and only if G is finite and e ( G ) = 2 if and only if G is virtually Z . Stallings’ celebrated theorem from [19] classifies finitely generated groups This research was partially supported by EPSRC grant EP/F031947/1. for which e ( G ) ≥
2. We state it here in its most general form as proved byDicks and Dunwoody using the Almost Stability Theorem.
Theorem 2. (Theorem IV.6.10 of [1] ) Let G be a group. The following areequivalent: (1) e ( G ) > H ( G, M ) = 0 , for any free G module M , (3) There exists a G -tree with finite edge stabilizers such that no vertexis stabilized by G . (4) One of the following holds: • G = B ∗ C D where B = C = D and C is finite, • G = B ∗ C , where C is finite, • G is countably infinite and locally finite. (5) the group G has 2 or infinitely many ends. The quest for a generalisation of this result covering splittings over arbi-trary subgroups has played a central role in low dimensional topology andgeometric group theory. The classical and algebraic annulus and torus the-orems are key examples (See [18] and references therein). While working onthis problem, Scott introduced in [16] an invariant e ( G, H ) for a subgroup H of a group G , which, in the case when G is finitely generated, can beidentified with the number of Freudenthal-Hopf ends of the quotient of alocally finite Cayley graph for G by the action of H . As with the classicalend invariant, e ( G, H ) does not depend on the choice of Cayley graph, andindeed the definition may be extended to the class of all discrete groups. Wepostpone the definition to section 1.Scott showed in [16] that if G splits as a non-trivial amalgamated freeproduct G = A ∗ C B or as an HNN extension G = A ∗ C then e ( G, H ) ≥ e ( G, { } ) = e ( G ) = e ( G, C ) for any finite subgroup
C < G ,Scott reformulated Stallings’ theorem as the statement that G splits over afinite subgroup if and only if e ( G, C ) ≥ C < G .He asked for which subgroups
H < G the analogous statement is true,remarking that it is certainly not true in general. For example the trianglegroup G = h a, b, c | a = b = c = ( ab ) = ( bc ) = ( ca ) i has an infinitecyclic subgroup H = h ab − i with e ( G, H ) = 2, but the group G does notsplit as an HNN extension, nor as a non-trivial amalgamated free product,over any subgroup. Scott’s resolution to this was the observation that while G does not split, it has a finite index subgroup G ′ which splits as an HNNextension over H .A more complete answer was given by the algebraic annulus theoremwhich asserts that if G is a one ended finitely generated group containinga two-ended subgroup H with e ( G, H ) ≥ G is virtually Z or G contains a two ended subgroup K over which it splits, or G has a finitenormal subgroup N whose factor group is a surface group. Here, we see twoways in which the obstruction to splitting over a subgroup can be overcome:one is to replace the group G by a finite index subgroup, the other is to adjust ELATIVE ENDS, ℓ -INVARIANTS AND PROPERTY (T) 3 the subgroup H . Both strategies play an important role in low dimensionaltopology. The latter is crucial in the statement and proof of the classicaltorus theorem (the fore-runner of the algebraic annulus and torus theorems)while the former is related to the virtual Haken and virtually positive firstBetti number conjectures.Scott’s proof that the triangle group contains a finite index subgroupwhich splits over the infinite cyclic subgroup H relied on the observation thatthe subgroup H is an intersection of finite index subgroups. Scott generalisedthis in [17] to show that if G is a finitely generated group, and H < G is afinitely generated subgroup which is an intersection of finite index subgroupsand such that e ( G, H ) ≥ G has a finite index subgroup which splitsover H . In particular, if G is a LERF group (i.e., a group in which everyfinitely generated subgroup is an intersection of finite index subgroups of G ), then every finitely generated subgroup H with e ( G, H ) ≥ G . Essentially the ideais that the obstruction to splitting G over H (sometimes referred to as thesingularity obstruction) is carried by finitely many double cosets of H in G and that by passing to a suitable finite index subgroup one removes all theseelements.In [8] the singularity obstruction S = Sing( G, H ) was studied in moredepth and it was shown that if
S ∪ H is contained in a proper subgroup G ′ of G then G will split over a subgroup of the group hS ∪ H i , whileif S is contained in the commensurator of H in G then G will split over asubgroup commensurable with H . Scott’s technique of passing to finite indexsubgroups was also strengthened to show that if the singularity obstructionis supported on n double cosets of H in G and H is contained in a strictlydecreasing chain of finite index subgroups of G of length at least n then G has a finite index subgroup which splits.While this last result has the advantage that it no longer requires H to bean intersection of finite index subgroups, the length of the chain required toensure that G virtually splits depends crucially on the size of the splittingobstruction and therefore, on the embedding of H in G . In an effort tocircumvent this difficulty we offer the following result (Corollary to Theorem4) which replaces the size of the singularity obstruction in the statement bya number which depends on the ℓ Betti numbers of H and G instead. Thishas the advantage that it is intrinsic to the groups H and G and does notdepend on the embedding of H in G , but comes with the disadvantage ofapplying only when G has positive ℓ Betti number, β (2)1 ( G ). See [13] forexamples. Corollary 3.
Let H ≤ G be discrete and countable one-ended groups suchthat β (2)1 ( G ) > . If ˜ e ( G, H ) ≥ and H is contained in a finite indexsubgroup G ′ < G with [ G : G ′ ] > β (2)1 ( H ) /β (2)1 ( G ) , then G ′ splits over asubgroup of the almost malnormal closure of H . (See Definition 5.) ADITI KAR AND GRAHAM A. NIBLO
The end invariant ˜ e ( G, H ) mentioned above is a generalisation of Scott’send invariant and was introduced by Kropholler and Roller, [7], in theirstudy of the algebraic torus theorem for Poincar´e duality groups. We willstate the definition of ˜ e ( G, H ) in section 1, but note here that in particularif e ( G, H ) ≥ e ( G, H ) ≥ ℓ cohomology, we refer the readerto [2]. Corollary 3 follows directly from Theorem 4 below. Note that groupswith non-trivial first ℓ betti number are either one-ended or have infinitelymany ends. In the latter case, Theorem 2 says that the group splits over afinite subgroup or is locally finite. Theorem 4.
Let H ≤ G be discrete and countable one-ended groups suchthat β (2)1 ( G ) > β (2)1 ( H ) . If ˜ e ( G, H ) ≥ then G splits over a subgroup of thealmost malnormal closure of H . Coxeter Groups
We now provide explicit examples in which the hypothe-ses of Theorem 4 are satisfied using the theory of Coxeter groups. Nibloand Reeves have shown in [9] that every finitely generated Coxeter group W = W ( S ) acts properly discontinuously on a locally finite, finite dimen-sional CAT(0) cube complex X W . Sageev’s work on ends of group pairsthen implies that e ( W, H ) ≥ H < W and it iseasy to deduce that H is the centralizer of a reflection. If W is a Coxetergroup with β ( W ) = 0, then one can extract additional information aboutthe structure of W using Theorem 4 and Corollary 3.To start with, let W be the Coxeter group generated by the reflections s , . . . , s such that s commutes with each of s , s and s while the pairwiseproduct of s with each of s , s , s and s is of infinite order. The pairwiseproducts of the generators s , s and s are of order 3. The remainingpairwise products are finite but greater than 50. Then W is a one-endedCoxeter group whose first ℓ betti number is non-zero, as can be seen fromapplying Theorem 3.2 of [13].Nuida describes the centralizers of reflections in his paper [12] and fromhis work, one deduces that the centralizer C of the reflection s is precisely T (3 , , ×h s i . Here, T (3 , ,
3) is the triangle group obtained from the para-bolic subgroup generated by s , s and s . As explained earlier, e ( W, C ) ≥ C contains Z as a finite index subgroup and therefore β ( C ) = 0.Using the same strategy one can build a whole family of examples us-ing the hyperbolic triangle groups T ( p, q, r ), where p , q and r are positiveintegers satisfying p + q + r <
1. This time, take W n to be a Coxetergroup generated by n reflections, s , . . . , s n . As in the earlier example, thereflection s commutes with precisely 3 other reflections s , s and s whilethe product of s with each of s , . . . , s n has infinite order. For sake ofsimplicity, we set the order of all pairwise products not already specifiedto be n . As before the centralizer C ( s ) is precisely T ( p, q, r ) × h s i and ELATIVE ENDS, ℓ -INVARIANTS AND PROPERTY (T) 5 e ( W n , C ( s )) ≥
2. Using Theorem 3.2 of [13] again, we have β ( W ) ≥ n − − (cid:18)
32 + 1 p + 1 q + 1 r + 1 n (cid:18) n ( n − − ( n − (cid:19)(cid:19) Now β ( C ( s )) is one-half of β ( T ( p, q, r )). Let χ ( . ) denote the orbifoldEuler characteristic of a group. One computes that χ ( T ( p, q, r )) = 12 (cid:18) p + 1 q + 1 r − (cid:19) Moreover, β ( T ( p, q, r )) = − χ ( T ( p, q, r )). This is a consequence of Atiyah’sformula relating the ℓ -Euler characteristic to the orbifold Euler characteris-tic. But for Fuchsian groups and in particular triangle groups, the argumentmay be simplified. Every triangle group contains a surface subgroup of finiteindex. Suppose T ( p, q, r ) contains a surface subgroup H ∼ = π ( S g ) (here, g is the genus) of index k . From first principles, β ( H ) = − χ ( S g ). Now, both β ( . ) and χ ( . ) are multiplicative on indices hence β ( T ( p, q, r )) = kβ ( H ) = k ( − χ ( S g )) = − χ ( T ( p, q, r ))Given p , q and r , for β ( W ) > β ( C ( s )) to hold, we need12 (cid:18) n − n + 4 n (cid:19) − (cid:18) p + 1 q + 1 r (cid:19) > − χ ( T ( p, q, r ))In particular if n − > χ ( T ( p, q, r )) + 2 then β ( W n ) > β ( C ( s )).One may specialise to the well-known (2 , ,
7) triangle group, which con-tains the fundamental group of the Klein’s quartic (a surface of genus 3) asa subgroup of index 336. Since β ( T (2 , , , one can choose n to be8 and get a splitting of W over T (2 , , × Z / Z . This splitting may alsobe obtained from visual decompositions of Coxeter groups into amalgams.It is worth noting here that the proof of Theorem 4 applies in moregenerality. Definition 5.
We will say that a subgroup H of a group G is almost mal-normal if for every g / ∈ H , the intersection H ∩ H g is finite.The almost malnormal closure of a subgroup H < G is the intersection ofthe almost malnormal subgroups of G containing H . We have the following generalisation of [6, Theorem 4.9].
Theorem 6.
Let H ≤ G be one-ended groups such that ˜ e ( G, H ) ≥ . Ifthe almost malnormal closure K of H is not equal to G then G splits overa subgroup of K . The
Kropholler conjecture is a long standing conjecture of Kropholler andRoller from [7]. To read more about the current status of the conjecture,see [11]. We show that our techniques give further evidence towards theconjecture by verifying it for pairs of groups H ≤ G satisfying β ( G ) >β ( H ). This is the content of Proposition 12. ADITI KAR AND GRAHAM A. NIBLO
The main protagonists of our next theorem are Poincar´e duality groups.An introduction to the notion of Poincar´e duality may be found in [6]. Fun-damental groups of closed aspherical manifolds are Poincar´e duality groups.Whether the converse is true for finitely presented groups is the subjectof a well known conjecture. One can show that the only one-dimensionalPoincar´e duality group is Z . That all Poincar´e duality of dimension 2 aresurface groups is a deep theorem established by Bieri, Eckmann, Mullerand Linnell. For each n ≥
4, Bestvina-Brady groups provide examples ofPoincare duality groups which are not finitely presented and hence are notfundamental groups of closed aspherical manifolds.We provide the following splitting theorem for Poincar´e duality groupswhich may be viewed as an analogue of the torus theorem and which playsa central role in the topological superrigidity theorem established in [5].
Theorem 7.
Let G be a Poincar´e duality group of dimension n . Supposethat H is an ( n − -dimensional Poincar´e duality subgroup of G and that H has property (T). Then G splits over a subgroup commensurable with H . For example suppose that M is a closed aspherical manifold of dimension4 n + 1, n ≥ N is a quarternionic hyperbolic closed manifold ofdimension 4 n which admits a π -injective map into M . Since π ( N ) hasproperty ( T ) the theorem shows that π ( M ) is a non-trivial amalgam orHNN extension over a subgroup commensurable with π ( N ). Note that thepresence of a codimension one property (T) subgroup in Theorem 7 becomesan obstruction to the ambient group having property (T).The paper is organised as follows. In section 1 we expand on the formaldefinition of the two end invariants e ( G, H ) and ˜ e ( G, H ) alluded to above. Insection 2 we give the proof of Theorems 4 and 6 and discuss the Krophollerconjecture. In section 3 we deal with Poincar´e duality and establish Theorem7.
Acknowledgements
We are grateful to Peter Kropholler, Indira Chatterjiand Ashot Minasyan for their comments and suggestions.1.
Relative Ends
Throughout the paper we will denote the field of order two by F . Nowlet G be a group and H be a subgroup of G . Given an H module M onemay form a G module using the functors Hom H ( F [ G ] , ) and F [ G ] ⊗ H .More precisely, choosing a set S of right coset representatives for H ≤ G wehave Coind GH M := Hom H ( F [ G ] , M ) ∼ = Y g ∈ S M g
Ind GH M := F [ G ] ⊗ H M ∼ = M g ∈ S M g
Let P G denote the collection of all subsets of G . Then, P G is an F -vector space with respect to the operation of symmetric difference. One ELATIVE ENDS, ℓ -INVARIANTS AND PROPERTY (T) 7 checks that P G is also a G module. Moreover, P G ∼ = Coind G F . On theother hand F H ( G ) = { A ⊆ G : A ⊆ HF for some finite set F } is the F G -module Ind GH P H . Similarly the power set P ( H \ G ) of H \ G andthe collection of finite subsets of H \ G , written F ( H \ G ) are F [ G ] modules.In fact, P ( H \ G ) ∼ = Coind GH F and F ( H \ G ) ∼ = Ind GH F . Definition 8.
The elements of F H ( G ) are said to be H -finite and the ele-ments of ( F H ( G ) \P G ) G are called H -almost invariant sets. Definition 9.
The algebraic end invariant is defined as ˜ e ( G, H ) = dim F ( F H ( G ) \P G ) G while the geometric end invariant is defined as e ( G, H ) = dim F ( F ( H \ G )) \P ( H \ G )) G . We collect together the properties of the end invariants defined abovewhich we will later need. The interested reader may find more details in [7].
Proposition 10.
Let H ≤ K ≤ G be groups. Then the following hold. (1) e ( G,
1) = e ( G ) = ˜ e ( G, . (2) e ( G, H ) = 0 = ˜ e ( G, H ) if and only if H has finite index in G . (3) If H has infinite index then ˜ e ( G, H ) = 1 + dim F H ( G, F H ( G )) . (4) If K has infinite index then ˜ e ( G, H ) ≤ ˜ e ( G, K ) . (5) e ( G, H ) = e ( X ) , where X is the coset graph of G with respect to H . (6) e ( G, H ) ≤ ˜ e ( G, H ) . Note that the algebraic end invariant for a group with infinitely many endswith respect to any of its infinite index subgroups is infinite. For instance,if G is the non-abelian free group of rank 2 and G ′ denotes its commutatorsubgroup, then ˜ e ( G, G ′ ) = ∞ (whereas e ( G, G ′ ) = 2). Clearly, the algebraicend invariant gives useful information only about one-ended groups.2. Proof of Theorems 4 and 6
Theorem 4.
Let H ≤ G be discrete and countable one-ended groups suchthat β (2)1 ( G ) > β (2)1 ( H ) . If ˜ e ( G, H ) ≥ then G splits over a subgroup of thealmost malnormal closure of H in G . Peterson and Thom showed in [13] that if β (2)1 ( G ) > β (2)1 ( H ) for a tor-sion free discrete countable group G then there exists a proper malnormalsubgroup H ′ of G that contains H . If one drops the hypothesis that G istorsion free then the same argument shows that H ′ is almost malnormal (seeDefinition 5). So Theorem 4 follows directly from Theorem 6. Theorem 6.
Let H ≤ G be one-ended groups such that ˜ e ( G, H ) ≥ . Ifthe almost malnormal closure K of H is not equal to G then G splits overa subgroup of K . ADITI KAR AND GRAHAM A. NIBLO
Proof.
Let H ≤ G be one-ended groups such that H is contained in a properalmost malnormal subgroup of G .Set Σ = { K < G : H ≤ K and K is almost malnormal in G } . Let( K j ) j ∈ J be elements of Σ and suppose g / ∈ ∩ j ∈ J K j . Then g does not belongto K j for at least one j ∈ J . As K j is almost malnormal in G , K j ∩ K gj isfinite. Thus, ( ∩ K j ) ∩ ( ∩ K j ) g is finite. We conclude that any intersection ofelements of Σ is almost malnormal and that Σ has a minimal element, thealmost malnormal closure of H which we will denote K . We will now showthat ˜ e ( G, K ) ≥ e ( K ) = 1.As the subgroup K is almost malnormal in G and G is infinite, K hasinfinite index in G . As noted in Proposition 10 the algebraic end invariant˜ e ( G, . ) is monotonic for infinite index subgroups, thus ˜ e ( G, K ) ≥ ˜ e ( G, H )and ˜ e ( G, K ) ≥ H in K limits the possibilities forthe value of e ( K ). Firstly K is infinite and so e ( K ) = 0. A group hastwo ends if and only if it is virtually Z . As K has a subgroup which is notvirtually Z , e ( K ) = 2. Thus K is either one ended or K has infinitely manyends. The latter is not a possibility, as we will now show.Suppose that K has infinitely many ends. Then by Theorem 2, K actson a tree T with no global fixed point and so that edge stabilisers are finite.We may restrict the action to H , but since e ( H ) = 1 this action does havea fixed point, and since H is infinite it cannot fix an edge so it must have afixed vertex. So H < A = Stab G ( v ) for some vertex v . We will show that A is almost malnormal in G . As K is minimal amongst the almost malnormalsubgroups containing H , this will imply that K < A and hence, A = K which contradicts the fact that K acts with no global fixed point on T .Suppose first that k ∈ K \ A . Then kv = v so A ∩ A k stabilises each edgeon the non-trivial geodesic from v to kv . It follows that A ∩ A k is finite.This tells us what happens for elements of G that lie in K . If g ∈ G \ K ,then K ∩ K g is finite and hence A ∩ A g which is contained in K ∩ K g isfinite. Thus A is almost malnormal in G .We now need to check that there exists a proper K almost invariant subset A in G such that AK = A . We generalise Kropholler’s methods in [6] todeal with the almost malnormal subgroups. The strategy will be to showthat for our choice of K , H ( K, F K ( G )) = 0. Recall that K is a one endedalmost malnormal subgroup of G such that ˜ e ( G, K ) ≥ K in G . As a K module, the induced module F K ( G ) is given byRes GK Ind GK P K ∼ = ⊕ g ∈ Λ Ind KK ∩ K g Res K g K ∩ K g P Kg.
The module Res K g K ∩ K g P Kg may be identified with Res KK g − ∩ K P K . Now, let g represent a non-trivial double coset of K in G . Then, we haveRes KK ∩ K g P K ∼ = Res KK ∩ K g Coind K F ∼ = Y ( K ∩ K g ) \ K Coind K ∩ K g F ℓ -INVARIANTS AND PROPERTY (T) 9 The subgroup K ∩ K g is finite and so the module Coind K ∩ K g F is isomorphicto the module Ind K ∩ K g F , which is precisely the group algebra F [ K ∩ K g ].Let R denote the algebra F [ K ∩ K g ]. Since R is finite, for any index set I , R I := Y I R ∼ = R ⊗ F I . To see this, observe that R I is the algebra of all R valued maps on I . Forany f : I → R and r ∈ R , define F ( r ) to be the set { i ∈ I : f ( i ) = r } . Thenthe assignment f X r ∈ R r ⊗ F ( r )is the required isomorphism. We deduce from this discussion that R I isa free module over the F -group algebra and it follows that P Kg is a free K ∩ K g -module. A module induced from a free module is also free and so wefind that F K ( G ) is the direct sum of P K and a free module. By Shapiro’sLemma, H ( K, P K ) = 0 for all groups K . Moreover, by Theorem 2, thefirst cohomology group of the one ended group K with respect to any freemodule is trivial. Thus, H ( K, F K ( G )) is zero.If B is a proper K almost invariant subset of G and H ( K, F K ( G )) iszero, then the derivation B B + Bg restricts to a principal derivation on K . There exists then a K -finite subset C such that B + Bx = C + Cx forall x ∈ K . Choose A to be B + C .Observe that for all g ∈ G \ K , ˜ e ( G, K ∩ K g ) = 1. This is because G isone ended and each of the intersections K ∩ K g is finite. The theorem nowfollows directly from Theorem 5.3 of [6]. (cid:3) A conjecture of Kropholler and Roller.
In the proof of Theorem4 we used the non-vanishing of the kernel of the restriction map
Res GH from H ( G, F K G ) to H ( H, F K G ) to extract a bi-invariant proper K almost in-variant subset of G and this in turn, helped to produce the splitting for thegroup. Kropholler and Roller conjectured the following: Conjecture 11. (Kropholler and Roller, [7] ) Let H ≤ G be finitely gener-ated groups. If G contains a proper H almost invariant subset A such that HAH = A , then G splits over a subgroup related to H . Here we provide further evidence in favour of the conjecture.
Proposition 12.
Conjecture 11 is true for all pairs G and H satisfying thehypotheses of the conjecture along with the condition β (2)1 ( G ) > β (2)1 ( H ) .Proof. The case when H is finite follows from Stallings’ celebrated Theoremon ends of groups. Assume that H is infinite. Then, as before, H is containedin a proper almost malnormal subgroup K of G .Choose A to be a proper H -almost invariant subset of G such that HAH = A and set S A ( G, H ) to be the set of elements g of the groupsuch that all four intersections A ∩ gA , A ∩ gA ∗ , A ∗ ∩ gA , and A ∗ ∩ gA ∗ are non-empty. This is the singularity obstruction defined in [8] and discussedabove.By Kropholler’s Lemma (4.17 of [6]), the condition that A = AH ensuresthat S A ( G, H ) is contained in the set S := { g ∈ G : ˜ e ( G, H ∩ H g ) ≥ } .Assume first that S is contained in K . Then, the singularity obstructionalong with the subgroup H generates a proper subgroup hS ∪ H i of G andthe main theorem of [8] asserts that G splits over a subgroup related to hS ∪ H i and hence to H . On the other hand, if S is not contained in K ,then for any g ∈ S\ K , ˜ e ( G, H ∩ H g ) ≥ H ∩ H g .Once again, by Stallings theorem on ends of groups, G splits over a subgroupcommensurable with H ∩ H g . This verifies the conjecture for our choice ofgroups G and H . (cid:3) Poincar´e duality groups
Theorem 7.
Let G be a Poincar´e duality group of dimension n . Supposethat H is an ( n − -dimensional Poincar´e duality subgroup of G and that H has property (T). Then G splits over a subgroup commensurable with H . An n -dimensional Poincar´e duality group is also called a P D n group. Proof.
Let G and H be as in the statement of the theorem. Then a simplecomputation shows that the end invariant ˜ e ( G, H ) is precisely 2. We includethe computation here for sake of completeness. Recall that ˜ e ( G, H ) = 1 +dim H ( G, F H ( G ). Denote the dualizing module H n ( G, F G ) by D G . Inour case, D G ∼ = F . Since G is a P D n group, we have H ( G, F H ( G )) ∼ = H n − ( G, Ind GH ( P H ⊗ F D G )). By Shapiro’s Lemma, H n − ( G, Ind GH ( P H ) ⊗ F D G )) ∼ = H n − ( H, P H ⊗ F D G ). Since H is a P D n − group, H n − ( H, P H ⊗ F D G ) is isomorphic to Hom F H ( D H , P H ⊗ F D G ) ∼ = F . Hence, ˜ e ( G, H ) = 2.We now invoke Lemma 2.5 of [7] to get a subgroup H ′ of finite index in H such that e ( G, H ′ )= ˜ e ( G, H )=2.Applying Sageev’s construction (see [14]) we obtain a CAT(0) cube com-plex X such that G acts essentially on X and H ′ is the stabilizer of anoriented codimension 1 hyperplane J . As H ′ has finite index in the prop-erty (T) group H , H ′ also has property (T). However, every action of agroup with property (T) on a CAT(0) cube complex must have a fixed point(see [10]) and so the action of H ′ on the CAT(0) cube complex J has aglobal fixed point. Hence, Lemma 2.5 from [15] implies the existence of aproper H ′ almost invariant subset B of G such that H ′ BH ′ = B .Recall that the singularity obstruction S B ( G, H ′ ) satisfies the following:for all g ∈ S B ( G, H ′ ), the subgroup K g defined as H ′ ∩ gH ′ g − has a properalmost invariant set B g such that K g B g = B g . But this implies that e ( G, K g )is at least 2.Every subgroup of infinite index in an n -dimensional Poincar´e Dualitygroup has cohomological dimension strictly less than n (See [20]). Moreover,for any P D n group X with subgroup Y of type FP, cd F Y ≤ n − e ( X, Y ) = 1 (Lemma 5.1 of [6]). This implies that K g has finite ELATIVE ENDS, ℓ -INVARIANTS AND PROPERTY (T) 11 index in both H ′ and gH ′ g − . More precisely, g lies in the commensuratorComm G ( H ′ ) of H ′ and S B ( G, H ) is a subset of Comm G ( H ′ ). Therefore byTheorem B of [8], G splits over a subgroup commensurable with H ′ . Thisproves the theorem. (cid:3) References [1] W. Dicks and M. Dunwoody, Groups acting on Graphs, Cambridge Studies in Ad-vanced Mathematics 17, CUP (1989).[2] B. Eckmann, Introduction to l -methods in topology : reduced l -homology, har-monic chains, l -Betti numbers. Notes prepared by Guido Mislin. Israel J. Math.117 (2000), 183–219.[3] H. Freudenthal, Uber die Enden topologischer Rume und Gruppen, Math. Z. (33),1 (1931) 692-713.[4] H. Hopf, Enden offener Raume und unendliche diskontinuierliche Gruppen, Com-ment. Math. Helvetici, 16 (1944), 81-100.[5] A. Kar and G. A. Niblo, Topological Superrigidity, Preprint 2011.[6] P. Kropholler, A Group Theoretic Proof of the Torus Theorem, Geometric GroupTheory Volume 1, Eds. G.A. Niblo and M. Roller, LMS Lecture Notes Series 181.[7] P. Kropholler and M. A. Roller, Relative Ends and Duality Groups, Journal of Pureand Applied Algebra, 61 (1989) 197-210, North Holland.[8] G. A. Niblo, The singularity obstruction for group splittings, Topology Appl. 119(2002), no.1, 17-31.[9] G. A. Niblo and L. D. Reeves, Coxeter groups act on CAT(0) cube complexes, J.Group Theory 6 (2003), 399-413.[10] G. A. Niblo and M. A. Roller, Groups acting on cubes and Kazhdan’s Property (T),Proc. Amer. Math. Soc., 126 (1998) 693-699.[11] G. A. Niblo and M. Sageev, The Kropholler Conjecture, L’Enseignement Mathma-tique (2) 54 (2008), 147-149.[12] K. Nuida, On reflections in Coxeter groups commuting with a reflection,arXiv:math/0603667v1.[13] J. Peterson and A. Thom, Group Cocycles and the ring of Affiliated Operators,arXiv:0708.4327v1.[14] M. Sageev, Ends of group pairs and non-positively curved cube complexes, Proc.London Math. Soc. 71 (1995), 585-617.[15] M. Sageev, Co-dimension 1 subgroups and splittings of groups, Journal of Algebra,189 (1997), 377-389.[16] G. P. Scott, Ends of pairs of groups, J. Pure Appl. Algebra, 11 (1977), 179-198.[17] G. P. Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc(2) 17 (1978), 555-565.[18] G. P. Scott, A new proof of the Annulus and Torus Theorems, Amer. J. Math., 102(1980) 241-277.[19] J. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88(1968) 312-334.[20] R. Strebel, A remark on subgroups of infinite index in Poincar´e duality groups,Comment. Math. Helv. 52(1977) 317-324. School of Mathematics, University of Southampton, Highfield, Southamp-ton, SO17 1SH, England
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