On the equivariant K - and KO -homology of some special linear groups
aa r X i v : . [ m a t h . K T ] A p r On the equivariant K - and K O -homology of some special linear groups
Sam HughesSchool of Mathematical Sciences, University of Southampton [email protected]
Abstract
We compute the equivariant KO -homology of the classifying space for proper actions of SL ( Z )and GL ( Z ). We also compute the Bredon homology and equivariant K -homology of the classifyingspaces for proper actions of PSL ( Z [ p ]) and SL ( Z [ p ]) for each prime p . Finally, we prove the unstableGromov-Lawson-Rosenberg conjecture for PSL ( Z [ p ]) when p ≡
11 (mod 12).
There has been considerable interest in the Baum-Connes conjecture; which states that for a group Γa certain ‘assembly map’, from the equivariant K -homology of the classifying space for proper actionsEΓ to the topological K -theory of the reduced group C ∗ -algebra, is an isomorphism. The Baum-Connesconjecture is known to hold for several families of groups, including word-hyperbolic groups, CAT(0)-cubical groups and groups with the Haagerup property. Conjecture 1.1 (The Baum-Connes Conjecture) . Let Γ be a discrete group, then the following a ssemblymap is an isomorphism µ : K Γ ∗ (EΓ) → K top ∗ ( C ∗ r (Γ)) . There is also a ‘real’ Baum-Connes conjecture which predicts that an assembly map from the equiv-ariant KO -homology of EΓ to the topological K of the real group C ∗ -algebra is an isomorphism. It isknown that the two conjectures are equivalent, that is, one of the assembly maps is an isomorphism ifand only if the other is [2]. Conjecture 1.2 (The Real Baum-Connes Conjecture) . Let Γ be a discrete group, then the following a ssembly map is an isomorphism µ R : KO Γ ∗ (EΓ) → K top ∗ ( C ∗ r (Γ; R )) . In spite of the interest, to date there have been very few computations of K Γ - and KO Γ -homology.Indeed, on the K Γ -homology side of things there are complete calculations for one relator groups [20],NEC groups [18], some Bianchi groups and hyperbolic reflection groups [16, 21, 22], some Coxeter groups[9, 27, 28], and SL ( Z ) [26]. For KO Γ -homology the author is aware of two complete computations; thefirst, due to Davis and L¨uck, on a family of Euclidean crystallographic groups [7], and the second, dueto Mario Fuentes-Rum´ı, on simply connected graphs of cyclic groups of odd order and of some Coxetergroups [10].In this paper we compute the topological side of the (real) Baum-Connes conjecture for the S -arithmetic lattices SL ( Z [ p ]), for p a prime, and for the arithmetic group SL ( Z ). In particular, we1ompute the equivariant K -homology of SL ( Z [ p ]) and the equivariant KO -homology of SL ( Z ). Wegive the relevant background and the connection to Bredon homology in Section 2. The calculation for KO Γ -homology is of particular interest because it is (to the author’s knowledge) the first computation of KO Γ ∗ for a property (T) group.This interest stems from the fact that property (T) is a strong negation of the Haagerup property.Moreover, the (real) Baum-Connes conjecture is still open for SL n ( Z ) when n ≥
3. We note that thereare counterexamples for the Baum-Connes conjecture for groupoids constructed from SL ( Z ) and moregenerally a discrete group with property (T) for which the assembly map is known to be injective [12]. Theorem 1.3.
Let
Γ = SL ( Z ) , then for n = 0 , . . . , we have KO Γ − n (EΓ) = Z , Z , Z , , Z , , , and the remaining groups are given by -fold Bott-periodicity. Applying a K¨unneth type theorem to the isomorphism GL ( Z ) ∼ = SL ( Z ) × Z on the level of Bredonhomology, we obtain the following result for GL ( Z ). Corollary 1.4.
Let
Γ = GL ( Z ) , then for n = 0 , . . . , we have KO Γ − n (EΓ) = Z , Z , Z , , Z , , , and the remaining groups are given by -fold Bott-periodicity. Moving onto Γ = PSL ( Z [ p ]) or SL ( Z [ p ]), for p a prime, we compute the equivariant K -homologygroups K Γ n (EΓ). There has been considerable interest in determining homological properties of the groupsSL ( Z [ m ]) and groups related to them [1, 4, 13]. It appears, however, that even with computer basedmethods the problem of determining the cohomology of SL ( Z [ m ]) for m a product of 3 primes is out ofreach [4]. In Lemma 5.3 we give a short proof of the Baum-Connes conjecture for SL ( Z [ p ]) and so weobtain the topological K -theory of the reduced group C ∗ -algebra of SL ( Z [ p ]) as well. Theorem 1.5.
Let p be a prime and Γ = PSL ( Z [ p ]) , then K Γ n (EΓ) is a free abelian group with rank asgiven in Table 1. Moreover, since the Baum-Connes and Bost conjectures hold for Γ we have K Γ ∗ (EΓ) ∼ = K top ∗ ( C ∗ r (Γ)) ∼ = K top ∗ ( ℓ (Γ)) . p = 2 p = 3 p ≡ p ≡ p ≡ p ≡
11 (mod 12) n = 0 7 6 4 + ( p −
7) 6 + ( p + 1) 5 + ( p −
1) 7 + ( p + 7) n = 1 0 0 3 1 2 0Table 1: Z -rank of the equivariant K -homology of the classifying space for proper actions of PSL ( Z [ p ]). Corollary 1.6.
Let p be a prime and Γ = SL ( Z [ p ]) then K Γ n (EΓ) is additively isomorphic to the directsum of two copies of the equivariant K -homology of PSL ( Z [ p ]) . Finally, we will give a proof of the unstable Gromov-Lawson-Rosenberg conjecture for positive scalarcurvature for the groups PSL ( Z [ p ]) when p ≡
11 (mod 12). The statement and background concerningthis conjecture is given in Section 6.
Theorem 1.7.
The unstable Gromov-Lawson-Rosenberg conjecture holds for the group
PSL ( Z [ p ]) when p ≡
11 (mod 12) . .1 Acknowledgements This paper contains contents from the author’s PhD thesis. The author would like to thank his PhDsupervisor Professor Ian Leary for his guidance and support. He would also like to thank Naomi Andrew,Guy Boyde, and Kevin Li for helpful conversations.
In this section we introduce the relevant background from Bredon homology and its interactions withequivariant K - and KO -homology. We follow the treatment given in Mislin’s notes [20]. Let Γ be a discrete group. A Γ-CW complex X is a CW-complex equipped with a cellular Γ-action. Wesay the Γ action is proper if all the cell stabilisers are finite.Let F be a family of subgroups of Γ which is closed under conjugation and finite intersections. A model for the classifying space E F Γ for the family F is a Γ-CW complex such that all cell stabilisersare in F and the fixed point set of every H ∈ F is weakly-contractible. This is equivalent the followinguniversal property; for every Γ-CW complex Y there is exactly one Γ-map Y → E F Γ up to Γ-homotopy.In the case where F = F IN (Γ), the family of all finite subgroups of Γ, we denote E FIN (Γ) by EΓ.We call such a space, the classifying space for proper actions of Γ. Note that if Γ is torsion-free thenEΓ = E Γ. Let Γ be a discrete group and F be a family of subgroups. We define the orbit category Or F (Γ) to be thecatgeory with objects given by left cosets Γ /H for H ∈ F and morphisms the Γ-maps φ : Γ /H → Γ /K .A morphism in the orbit category is uniquely determined by its image ϕ ( H ) = γK and γHγ − ⊆ K ;conversely, each such γ ∈ Γ defines a G -map.A (left) Bredon module is a covariant functor M : Or F (Γ) → Ab , where Ab is the category of Abeliangroups. Consider a Γ-CW complex X and a family of subgroups F containing all cell stabilisers. Let M be a Bredon module and define the Bredon chain complex with coefficients in M as follows:Let { c α } be a set of orbit representatives of the n -cells in X and let Γ α denote the stabiliser of thecell α . The n th chain group is then C n := M α M (Γ / Γ c α ) . If γc ′ is an ( n − c , then γ − Γ c γ ⊆ Γ c ′ yielding a Γ-map ϕ : Γ / Γ c → Γ / Γ c ′ . Thisgives a induced homomorphism M ( ϕ ) : M (Γ / Γ c ) → M (Γ / Γ c ′ ). Putting this together we obtain a differ-ential ∂ : C n → C n − . Taking homology of the chain complex ( C ∗ , ∂ ) gives the Bredon homology groups H F n ( X ; M ). A right Bredon module and Bredon cohomology is defined analogously with contravariantfunctors. K -homology Let Γ be a discrete group. In the context of the Baum-Connes conjecture we are specifically interestedin the case where X = EΓ, F = F IN (Γ) and M = R C the complex representation ring. We consider3 C ( − ) as a Bredon module in the following way, for Γ /H ∈ Or F (Γ) set R C (Γ /H ) = R C ( H ), the ring ofcomplex representations of the finite group H . Morphisms are then given by induction of representations.We note that R C (Γ) := H F (Γ) = colim Γ /H ∈ Or (Γ) R C ( H ). In the case that Γ has finitely manyconjugacy classes of finite subgroups, R C (Γ) is a finitely generated quotient of L R C ( H ), where H runsover conjugacy classes of finite subgroups.Of course we still have to show how Bredon homology links to K Γ ∗ (EΓ), the equivariant K -homologyof the classifying space for proper actions. The answer is provided by Kasparov’s KK -theory and thefollowing equivariant Atiyah-Hirzebruch type spectral sequence.More specifically for each subgroup H ≤ Γ we have K Γ n (Γ /H ) = K i ( C ∗ r ( H )). In the case H is afinite subgroup then C ∗ r ( H ) = C H , K Γ0 (Γ /H ) = K ( C H ) = R C ( H ), and K Γ1 (Γ /H ) = K ( C H ) = 0. Theremaining K Γ groups are given by 2-fold Bott periodicity. Theorem 2.1. [20, Page 50]
Let Γ be a group and X a proper Γ -CW complex, then there is an Atiyah-Hirzebruch type spectral sequence E p,q := H FIN p ( X ; K Γ q ( − )) ⇒ K Γ p + q ( X ) . KO -homology Again fixing a discrete group Γ and F = F IN (Γ), we introduce two more Bredon modules, the realrepresentation ring R R ( − ), and the quaternionic representation ring R H ( − ). These are defined on Or F (Γ)in exactly the same way as the complex representation ring. We have natural transformations betweenthe functors. Indeed for a finite subgroup H ≤ Γ we have a diagram (which does not commute): R R ( H ) R C ( H ) R H ( H ) . ν σρ η For a real representation ψ , the complexification is ν ( ψ ) = ψ ⊗ C . For a complex representation φ , the symplectification is σ ( φ ) = φ ⊗ H . Going the other way, for an n -dimensional quaternionic representation ω ,the complexification is η ( ω ) = η considered as 2 n complex representation. Similarly, for an n -dimensionalcomplex representation φ , the realification is ρ ( φ ) = φ considered as a 2 n -dimensional real representation.Note that any composition of the x-ification natural transformations with the same source and target isnecessarily not the identity.The situation for the equivariant KO -homology, denoted KO Γ ∗ ( − ), is similar to the equivariant K -homology but more complicated. For a subgroup H ≤ Γ we set KO Γ n (Γ /H ) = KO n ( C ∗ r ( H ; R )). In thecase that H is a finite subgroup we now have that KO Γ − n (Γ /H ) = KO − n ( C ∗ r ( H ; R )) = R R ( H ) n = 0 , R R ( H ) /ρ ( R C ( H ) n = 1 , R C ( H ) /η ( R H ( H )) n = 2 , n = 3 , R H ( H ) n = 4 , R H ( H ) /σ ( R C ( H )) n = 5 , R C ( H ) /ν ( R R ( H )) n = 6 , n = 7 , X a proper Γ-space, the Atiyah-Hirzebruchspectral sequence from before now takes the form E p,q := H FIN p ( X ; KO Γ q ( − )) ⇒ KO Γ p + q ( X ) . The section gives an alternative Γ-equivariant homotopy theoretic viewpoint. Now we consider Γ-equivariant homology theories as functors E : Or F (Γ) → Spectra . Technically, to avoid functorialproblems one must take composite functors through the categories C ∗ - Cat and
Groupoids . We donot concern ourselves with this complication and refer the reader to [5] and [8].Instead we will take for granted that there is a composite functor KO : Or F (Γ) → Spectra which satisfies π n KO (Γ /H ) = KO n ( C ∗ r ( H ; R )). When F = F IN this perspective gives a homotopytheoretic construction of the (real) Baum-Connes assembly map. Indeed, we have maps B Γ + ∧ KO ≃ hocolim Or T RV (Γ) KO → hocolim Or F (Γ) KO → hocolim Or ALL (Γ) KO ≃ KO ( C ∗ r (Γ; R )) . The assembly map µ R is then π n applied to the composite. K O -homology of SL ( Z ) A model for X = ESL ( Z ) can be constructed as a SL ( Z )-equivariant deformation retract of the sym-metric space SL ( R ) /O (3). This construction has been detailed several times in the literature ([31,Theorem 2], [11, Theorem 2.4] or [26, Theorem 13]), so rather than detailing it again here, we simplyextract the relevant cell complex and cell stabilisers. Specifically, we follow the notation of S´anchez-Garc´ıa[26] and collect the information in Table 2. The calculation of the equivariant KO -groups will follow from the following proposition and an analysisof the representation theory of the finite subgroups of SL ( Z ). We remark that one could prove a dozensubtle variations on the theme of the following proposition. However, rather than do this we introduce aslogan: “Computations with coefficients in KO Γ n ( − ) can be greatly simplified by looking for chain mapsto the Bredon chain complex with coefficients in R C ( − ).” Proposition 3.1.
Let Γ be a discrete group, F = F IN (Γ) and X = EΓ . Assume that for everycell stabiliser the real, complex and quarternionic character tables are equal, then the Atiyah-Hirzebruchspectral sequence converging to KO Γ ∗ (EΓ) has E -page isomorphic to E p,q = H FIN p (EΓ); K Γ0 ( − )) ⊗ KO q ( ∗ ) where for n = 0 , . . . , we have KO − n ( ∗ ) = Z , Z , Z , , Z , , , and the remaining groups are given by -fold Bott-periodicity. T − t + t − t + t − t { } t e − e − e Z t e − e + e { } t e − e + e Z t e − e + e + e Z t e − e + e − e + e Z e v − v Z e v − v D e v − v D e v − v Z e v − v Z e v − v Z e v − v D e v − v D v - Sym(4) v D v Sym(4) v D v Sym(4)Table 2: Cell structure and stabilisers of a model for ESL ( Z ). Proof.
First note that when n = 3 or 7 the result is immediate since KO Γ n ( − ) = 0. Now, since the threecharacter tables are equal the complexification from R R ( − ) to R C ( − ) and the symplectification from R C ( − ) to R H ( − ) are isomorphisms. In particular, KO Γ5 ( − ) = R H ( − ) /σ ( R C ( − )) = 0 and KO Γ6 ( − ) = R C ( − ) /ν ( R R ( − )) = 0. Moreover, it follows they induce isomorphisms on the Bredon chain complexes C ∗ ( X ; R R ) ∼ = C ∗ ( X ; R C ) ∼ = C ∗ ( X ; R H ). In particular, KO Γ0 ( − ) ∼ = KO Γ4 ( − ) ∼ = K Γ0 ( − ). Finally, thecomplexification from R H ( − ) to R C ( − ) and the realification from R C to R R correspond to multiplicationby 2. Thus, when we take cokernels and pass to the Bredon chain complex for the bredon modules KO Γ1 ( − ) and KO Γ2 ( − ), we obtain an isomorphism to the modulo 2 reduction of Bredon chain complexfor K Γ0 ( − ). Proof of Theorem 1.3.
Let Γ = SL ( Z ), F = F IN (Γ) and X = ESL ( Z ). We can now complete thecalculation for the equivariant KO -homology groups. First, we recap calculation of the Bredon chaincomplex with complex representation ring coefficients due to S´anchez-Garc´ıa. We have a chain complex0 Z Z Z Z ∂ ∂ ∂ where ∂ ∼ (cid:2) × (cid:3) , ∂ ∼ (cid:20) I × × × (cid:21) , and ∂ ∼ (cid:20) I × × × (cid:21) . Therefore, the homology groups of the chain complex are isomorphic to Z in dimension 0 and to 0 inevery other dimension. 6ow, the cell stabiliser subgroups of SL ( Z ) acting on X are isomorphic to { } , Z , Z , D , D ,Sym(4) and D . Each of which satisfies the conditions of the proposition above. Applying this to theprevious calculation we obtain a single non-trivial column when p = 0 in the Atiyah-Hirzebruch spectralsequence and so it collapses trivially. Proof of Corollary 1.4.
The result for GL ( Z ) follows from the observation that the direct product of Z with any of the cell stabiliser subgroups still satisfies the conditions of the Proposition 3.1 and so we haveisomorphisms KO GL ( Z ) n (EGL ( Z )) ∼ = KO SL ( Z ) n (ESL ( Z )) ⊗ KO Z n ( ∗ ) . K -homology of Fuchsian Groups In this section we compute the equivariant K -homology of every finitely generated Fuchsian group, thatis a finitely generated discrete subgroup of PSL ( R ). Note that Theorem 4.1(a) was computed in [18]along with a more general result for cocompact NEC groups. Moreover, their integral cohomology wasdetermined by the author in [13]. Finally, we remark that the equivariant K -homology for orientable(and some non-orientable) NEC groups can be easily determined via an appropriate graph of groupsconstruction. However, we leave this as an exercise for the interested reader.The reason for this apparent detour is that we will later split the groups PSL ( Z [ p ]) as amalgamatedfree products of certain Fuchsian subgroups. Thus, we can use a Mayer-Vietoris type argument to computetheir K -homology. This is made easier by the fact that every finitely generated Fuchsian group is describedby piece of combinatorial data called a signature . Indeed, a Fuchsian group of signature [ g, s ; m , . . . , m r ]has presentation * a , b , . . . , a g , b g , c , . . . , c s , d . . . . , d r | g Y i =1 [ a i , b i ] r Y j =1 d j s Y k =1 c k = d m = · · · = d m r r = 1 + and acts on the hyperbolic plane R H with a 4 g + 2 s + 2 r sided fundamental polygon. The tessellationof the polygon under the group action has 1 + s + r orbits of vertices, s of which are on the boundary ∂ R H , 2 g + s + r orbits of edges and 1 orbit of faces. All edge and face stabilisers are trivial. All vertexstabilisers are trivial except for r orbits of vertices, each of which is stabilised by some Z m j . Note that if s = 0 we say Γ is cocompact. Theorem 4.1.
Let Γ be a Fuchsian group of signature [ g, s ; m , . . . , m r ] , then(a) if s = 0 , K Γ n (EΓ) = ( Z r + P rj =1 m j n even, Z g n odd.(b) if s > , K Γ n (EΓ) = ( Z r + P rj =1 m j n even, Z g + s − n odd. roof of (a). Let Γ be a Fuchsian group of signature [ g, s ; m , . . . , m r ] with s = 0. The hyperbolic planewith the induced cell structure of the Γ action is a model for E G (see for instance [19]). Recall that thecell structure has r + 1 orbits of vertices, 2 g + r orbits of edges and exactly 1 orbit of 2-cells. One vertex v is stabilised by the trivial group and for j = 1 , . . . , r the vertex v j is stabilised y Z m j . Thus, we havea Bredon chain complex0 Z ⊕ (cid:16)L rj =1 R C ( Z m j ) (cid:17) Z g + r Z , ∂ ∂ substituting in R C ( Z m j ) = Z m j we obtain0 Z ⊕ (cid:16)L rj =1 Z m j (cid:17) Z g + r Z . ∂ ∂ We fix the following basis for each chain group: In degree 0 we have generators x j,l , for j = 1 , . . . , r and l = 1 , . . . , m j , and the generator z . In degree 1 we have a , b , . . . , a g , b g and y , . . . , y r , and in degree2, the generator w . A good stare reveals that ∂ ( w ) = 0, ∂ ( a i ) = ∂ ( b i ) = 0, and ∂ ( y j ) = P m j l =1 x j,l − z .Thus, H FIN n (EΓ; R C ) = Z P rj =1 ( m j − if n = 0; Z g if n = 1; Z if n = 2;0 otherwise . From here we apply the equivariant AHSS, since the homology in concentrated in degrees less than orequal to 2 there are no non-trivial differentials. Moreover, since every term is free abelian, the extensionproblems are resolved trivially. It follows that K Γ0 (EΓ) = H FIN (EΓ; R C ) ⊕ H FIN (EΓ; R C ) and K Γ1 (EΓ) = H FIN (EΓ; R C ). Z m Z m r g + s − Proof of (b).
Let Γ be a Fuchsian group of signature [ g, s ; m , . . . , m r ] with s >
0. In this case we canrearrange the presentation of Γ such that we have a splitting of Γ as an amalgamated free productΓ ∼ = Z s − ∗ Z m ∗ · · · ∗ Z m r . Now, Γ splits as a finite graph of finite groups (Figure 1) and it is easy to seethat the Bass-Serre tree of Γ is a model for EΓ.We first compute the Bredon homology H FIN∗ (EΓ; R C ) with coefficients in the representation ring,from here we can apply the equivariant Atiyah-Hirzebruch spectral sequence. For EΓ we have a Bredonchain complex 0 Z ⊕ (cid:16)L rj =1 R C ( Z m j ) (cid:17) Z g + s − , ∂ R C ( Z m j ) = Z m j we obtain0 Z ⊕ (cid:16)L rj =1 Z m j (cid:17) Z g + s − . ∂ Let the first non-zero term have generating set h x j,l , z | j = 1 , . . . , r, l = 1 , . . . , m j i and the secondterm h a , b . . . , a g , b g , c , . . . , c s − , d , . . . , d r i . It is easy to see the differential ∂ is given by ∂ ( a i ) = ∂ ( b i ) = ∂ ( c k ) = 0 and ∂ ( d j ) = P m j l =1 x j,l − z . It follows that H FIN (EΓ; R C ) = Z P rj =1 ( m j − , H FIN (EΓ; R C ) = Z g + s − and 0 otherwise. Since the Bredon homology is concentrated in degrees 0 and 1 it follows theequivariant AHSS collapses trivially. In particular, we have K Γ n (EΓ) = H F n (EΓ; R C ) for n = 0 , PSL ( Z [ p ]) and SL ( Z [ p ]) In an abuse of notation, throughout this section we will denote the image {± A } of matrix A ∈ SL ( Z [ p ])in PSL ( Z [ p ]) by the matrix A . Recall that for p a prime we have PSL ( Z [ p ]) = PSL ( Z ) ∗ Γ ( p ) PSL ( Z ),where Γ ( p ) is the level p Hecke principle congruence subgroup (see for instance Serre’s book “Trees”[29]). The amalgamation is specified by two embeddings of the congruence subgroup Γ ( p ) into PSL ( Z ).The first is given by Γ ( p ) := (cid:26)(cid:20) a bc d (cid:21) ∈ PSL ( Z ) : c ≡ p ) (cid:27) and the second via (cid:20) a bc d (cid:21) (cid:20) a pbp − c d (cid:21) . In light of this we will collect some facts about each of the groups in the amalgamation. We begin byrecording (Table 3) the Fuchsian signatures and the associated Bredon homology for each of the groupsΓ ( p ) and PSL ( Z ). p Signature of Γ ( p ) H F (EΓ ( p ); R C ) H F (EΓ ( p ); R C )2 [0 ,
2; 2] Z Z ,
2; 3] Z Z p ≡ , ( p −
7) + 1; 2 , , , Z Z ( p − p ≡ , ( p + 1) + 1; 2 , Z Z ( p +1) p ≡ , ( p −
1) + 1; 3 , Z Z ( p − p ≡
11 (mod 12) [0 , ( p + 7) + 1; ] Z Z ( p +7) PSL ( Z ) [0 ,
1; 2 , Z ( p ). Lemma 5.1.
The signatures and Bredon homology groups listed in Table 3 are the signatures and Bredonhomology groups of Γ ( p ) and PSL ( Z ) .Proof. The Bredon homology follows from the computation of the signatures and Theorem 4.1. Note thatthe fact the signature of PSL ( Z ) is [0 ,
1; 2 ,
3] is well known. To determine the signatures for the other9roups, we first note that it follows from [13] that two finitely generated Fuchsian groups are isomorphicif and only if their (co)homology groups are isomorphic. As Γ ( p ) is a subgroup of PSL ( Z ) = Z ∗ Z ,both the torsion of Γ ( p ) and the torsion in the cohomology of Γ ( p ) contains elements of order 2 or 3.Moreover, in their signature g = 0.Next, we consult the computations of the homology of principal congruence subgroups f Γ ( p ) of SL ( Z )in [1]. Now, consider the Lyndon-Hochschild-Serre spectral sequence for the group extension Z f Γ ( p ) ։ Γ ( p ) which takes the form E r,s = H r (Γ ( p ); H s ( Z ; Z )) ⇒ H r + s ( f Γ ( p ); Z ) . We immediately deduce that s = b ( f Γ ( p )) + 1 and the m j can be deduced from comparing the E -page,the results in [13] and the results in [1]. Indeed, each m j can only equal 2 or 3 and corresponds to a Z or Z summand in the cohomology of Γ ( p ).We shall also record the conjugacy classes of finite order elements of Γ ( p ) and PSL ( Z [ p ]). Note thatthe only conjugacy classes of finite subgroups of PSL ( Z ) are one class of groups isomorphic to Z andone to Z since PSL ( Z ) ∼ = Z ∗ Z . The conjugacy classes of finite subgroups of Γ ( p ) can be read off ofthe signature, there is exactly one of order m j for each j = 1 , . . . , r . Lemma 5.2.
The number of conjugacy classes of finite order elements in
PSL ( Z [ p ]) are those given inTable 4. p = 2 p = 3 p ≡ p ≡ p ≡ p ≡
11 (mod 12)Identity 1 1 1 1 1 1Order 2 1 2 1 1 2 2Order 3 4 2 2 4 2 4Total 6 5 4 6 5 7Table 4: Number of conjugacy classes of finite order elements of PSL ( Z [ p ]). Proof.
The result follows from the following observation: If there is a conjugacy of elements of order 2(resp. 3) in Γ ( p ), then each of class of elements of order 2 (resp. 3) in PSL ( Z ) fuses in PSL ( Z [ p ]). Tosee this, consider an element in the first copy of PSL ( Z ), conjugate it to an element in Γ ( p ), and thenconjugate it to an element in the other copy of PSL ( Z ). Lemma 5.3.
Both SL ( Z [ p ]) and PSL ( Z [ p ]) satisfy the Baum-Connes Conjecture.Proof. Since PSL ( Z [ p ]) = PSL ( Z ) ∗ Γ ( p ) PSL ( Z ), the Bass-Serre tree of the amalgamation is a locally-finite 1-dimensional contractible PSL ( Z [ p ])-CW complex. Moreover, each of the stabilisers Γ c havecd Q (Γ c ) = 1, being a graph of finite groups. Now, we apply [20, Corollary 5.14] to see the stabiliserssatisfy Baum-Connes and [20, Theorem 5.13] to see that PSL ( Z [ p ]) does. The proof is identical forSL ( Z [ p ]). 10 .2 Computations There is a long exact Mayer-Vietoris sequence for computing the Bredon homology of an amalgamatedfree product.
Theorem 5.4. [20, Corollary 3.32]
Let
Γ = H ∗ L K and let M be a Bredon module. There is a long exactMayer-Vietoris sequence: · · · H FIN n (E L × Γ Γ; M ) H FIN n (E H × Γ Γ; M ) ⊕ H FIN n (E K × Γ Γ; M ) · · · H FIN n − (E L × Γ Γ; M ) H FIN n (EΓ; M )We are now ready to compute the K -theory of PSL ( Z [ p ]). Proof of Theorem 1.5.
There are 6 cases to consider. Let Γ = PSL ( Z [ p ]) and R C = R C (Γ). In each casewe have the following long exact Mayer-Vietoris sequence0 H FIN (Γ; R C ) H FIN (EΓ × Γ Γ ( p ); R C ) (cid:0) H FIN (EΓ × Γ PSL ( Z ); R C ) (cid:1) H FIN (Γ; R C ) H FIN (EΓ × Γ Γ ( p ); R C ) (cid:0) H FIN (EΓ × Γ PSL ( Z ); R C ) (cid:1) H FIN (Γ; R C ) 0 . For Λ ≤ Γ we have H FIN n (EΓ × Γ Λ; R C (Γ)) ∼ = H FIN n (Λ; R C (Λ)). We also computed the Bredon homol-ogy groups of PSL ( Z ) and Γ ( p ) in Table 3. Thus, we can separate the above sequence into two sequences.Indeed, H FIN (PSL ( Z ); R C (PSL ( Z ))) = 0, so it follows that H FIN (Γ; R C ) ∼ = H FIN (Γ ( p ); R C (Γ ( p ))).The other sequence is then given by the remaining terms.We will treat the case p = 2, the other cases are identical. We have H FIN (Γ; R C ) = Z and an exactsequence 0 H FIN (Γ; R C ) Z Z H FIN (Γ; R C ) 0 . We now compute the colimit H FIN (Γ; R C ) = colim Γ /H ∈ Or (Γ) R C ( H ). Since we have a completedescription of the conjugacy classes of finite subgroups of Γ and the only inclusions are given by { } ֒ → Z and { } ֒ → Z , it follows that H FIN (Γ; R C ) = Z . Moreover, for the sequence to be exact, if follows themap Z → Z must be an isomorphism onto the kernel of the first map. In particular H FIN (Γ; R C ) = 0.The equivariant K -homology then follows from applying the equivariant Atiyah-Hirzebruch spectralsequence, where we find K Γ0 (EΓ) = H FIN (Γ; R C ) ⊕ H FIN (Γ; R C ) and K Γ0 (EΓ) = H FIN (Γ; R C ). Werecord the Bredon homology groups for the remaining cases in Table 5, the reader can easily verify these.Note that they are always torsion-free and so are completely determined by their Z -rank.To extend the calculations to SL ( Z [ p ]) we use the following proposition and then apply the equivariantAtiyah-Hirzebruch spectral sequence. 11 = 2 p = 3 p ≡ p ≡ p ≡ p ≡
11 (mod 12) n = 0 6 5 4 6 5 7 n = 1 0 0 3 1 2 0 n = 2 1 1 ( p − ( p + 1) ( p − ( p + 7)Table 5: Z -rank of the Bredon homology of PSL ( Z [ p ]). Proposition 5.5.
Let A Γ ։ Q be a short exact sequence of groups, where A is a central finite abeliangroup of order n . If each finite subgroup F of Γ is abelian, then H FIN∗ (Γ; R C ) = n M i =1 H FIN∗ ( Q ; R C ) . Proof.
Consider E Q as a Γ-space via the quotient map, clearly E Q is a model for EΓ. Let F be theimage of a cell-stabiliser F in Q , then F ∼ = F × A or F ∼ = F · A , where · denotes a non-split centralextension such that F is abelian. Passing to C Γ ∗ (EΓ; R C ), the Bredon chain complex with coefficients inthe representation ring, we see that each chain group is isomorphic to | A | copies of the correspondingchain group in C Q ∗ (E Q ; R C ).If each stabiliser splits as F ∼ = F × A , then result is immediate by the K¨unneth formula. If not thenthe problem reduces to a computation of induced representations of finite abelian groups. Indeed, theonly situation where problems may occur is if A and F both contain a Z n summand, such that F containsa Z n summand. Assume H is the stabiliser of a cell one dimension higher, such that H ≤ F , so A ≤ H .There are two cases, either the differential in C Q ∗ (E Q ; R C ) restricted to R C ( H ) is 0, and so the differentialin C Γ ∗ (EΓ; R C ) is also 0. Or, the differential restricts to an inclusion and so the differential in C Γ ∗ (EΓ; R C )is an inclusion as well. Given a smooth closed n -manifold M a classical question is to ask whether M admits a Riemannian metricof positive scalar curvature. In a vast generalisation of the Atiyah-Singer index theorem, Rosenberg [24]exhibits a class in KO n ( C ∗ r ( π ( M ) , R )) which is an obstruction to M admitting a metric of positive scalarcurvature.More precisely, let M be a closed spin n -manifold and f : M → B Γ be a continuous map for somediscrete group Γ. Let α : Ω Spin n ( B Γ) → KO n ( C ∗ r (Γ; R )) be the index of the Dirac operator. If M admitsa metric of positive scalar curvature, then α [ M, f ] = 0 ∈ KO n ( C ∗ r (Γ; R )) Conjecture 6.1 (The (unstable) GLR conjecture) . Let M be a closed spin n -manifold and Γ = π ( M ) .If f : M → B Γ is a continuous map which induces the identity on the fundamental groups, then M admitspositive scalar curvature if and on if α [ M, f ] = 0 ∈ KO n ( C ∗ r (Γ; R )) . The conjecture has been verified in the case of some finite groups [32, 25, 15, 23], when the group hasperiodic cohomology, torsion-free groups for which the dimension of B Γ is less than 9 [14], and cocompactFuchsian groups [8]. However, there are counterexamples, the first, isomorphic to a semi-direct product Z ⋊ Z , is due to Schick [30], however, other counterexamples have since been constructed [14].12 .1 Proof of Theorem 1.7 We will now prove the conjecture for Γ = PSL ( Z [ p ]). Our proof is structurally similar to the proof byDavis-Pearson [8] so we will summarise their method and highlight any differences.Let ko be the connective cover of KO with covering map p and let D be the ko -orientation of spinbordism. The map α (from above) is obtained by the following compositionΩ Spin n ( B Γ) ko n ( B Γ) KO n ( B Γ) KO n ( C ∗ r (Γ; R )) D p µ R We note that ko n ( ∗ ) = 0 for n < p is an isomorphism for n ≥ Proposition 6.2.
Let p ≡
11 (mod 12) be prime. Let
Γ = PSL ( Z [ p ]) and X = EΓ / Γ and F = F IN .Let Λ be a set of conjugacy classes of finite subgroups of Γ . There is a commutative diagram with exactrows g KO n +1 ( X ) L ( H ) ∈ Λ g KO n ( BH ) g KO n ( B Γ) g KO n ( X ) g KO n +1 ( X ) L ( H ) ∈ Λ g KO n ( C ∗ r ( H ; R ))) g KO n ( C ∗ r (Γ; R ))) g KO n ( X ) . id µ R µ R id Proof.
First, we observe that Γ = PSL ( Z [ p ]) satisfies the following two conditions:(M) Every finite subgroup is contained in a unique maximal finite subgroup.(NM) If M is a maximal finite subgroup, then the normaliser N Γ ( M ) of M is equal to M .These are both easily seen from the classification of finite subgroups and the amalgamated productdecomposition. Note that the failure of (NM) for the other primes is the main obstruction to generalisingthis proof. Now, by either [6, Corollary 3.13] or the proof of [6, Theorem 4.1] for any constant functor E c : Or F (Γ) → Spectra by Γ /H E there are long exact sequences · · · L ( H ) ∈ Λ H n ( BH ; E ) L ( H ) ∈ Λ π n ( E ) ! ⊕ H n ( B Γ; E ) H n ( X ; E ) · · · and · · · L ( H ) ∈ Λ e H n ( BH ; E ) e H n ( B Γ; E ) e H n ( X ; E ) · · · The result then follows a diagram chase exactly as in [8, Proposition 4], taking E = KO and the isomor-phism π n (cid:18) hocolim Or F (Γ) ( E c ) (cid:19) ∼ = H n ( X ; E ) . roof of Theorem 1.7. Let p ≡
11 (mod 12) be prime. Let Γ = PSL ( Z [ p ]) and X = EΓ / Γ. Let ko be thespectrum of the connective cover of KO . Via the cover we obtain a natural transformation p : ko c → KO c of constant Or F (Γ)- Spectra . From the previous proposition we obtain a commutative diagram f ko n +1 ( X ) L ( H ) ∈ Λ f ko n ( BH ) f ko n ( B Γ) f ko n ( X ) g KO n +1 ( X ) L ( H ) ∈ Λ g KO n ( C ∗ r ( H ; R ))) g KO n ( C ∗ r (Γ; R ))) g KO n ( X ) . p µ R ◦ p µ R ◦ p p ( † )We claim that Σ X = Σ(EΓ / Γ) ≃ W b (Γ ( p )) S , where b (Γ ( p )) is the first Betti number of Γ ( p ).Indeed, we have X ≃ hocolim Top (cid:0) (EPSL ( Z ) × PSL ( Z ) Γ) ← (EΓ ( p ) × Γ ( p ) Γ) → (EPSL ( Z ) × PSL ( Z ) Γ) (cid:1) / Γ , ≃ hocolim Top (cid:0) (EPSL ( Z ) × PSL ( Z ) Γ) / Γ ← (EΓ ( p ) × Γ ( p ) Γ) / Γ → (EPSL ( Z ) × PSL ( Z ) Γ) / Γ (cid:1) . Since EPSL ( Z ) / PSL ( Z ) is an interval and EΓ ( p ) / Γ ( p ) is a finite graph, we have X ≃ hocolim Top I ← _ b (Γ ( p )) S → I , but I is contractible, so up to homotopy this becomes a suspension of a wedge of circles. In particular, X ≃ _ b (Γ ( p )) S . It follows that f ko ( X ) ∼ = ko n − ( ∗ ) b (Γ ( p )) and g KO ( X ) ∼ = KO n − ( ∗ ) b (Γ ( p )) , therefore, the naturaltransformation p is an isomorphism for n ≥
2. Now, suppose that n ≥ β ∈ K := Ker ( µ R ◦ p : ko n ( B Γ) → KO n ( C ∗ r (Γ; R ))and note K ∼ = Ker( µ R ◦ p : f ko n ( B Γ) → g KO n ( C ∗ r (Γ; R )). Combining the diagram ( † ) with the isomorphism p : f ko n ( X ) → g KO n ( X ), we can deduce that there exists γ ∈ Ker M ( H ) ∈ Λ ko n ( BH ) → M ( H ) ∈ Λ KO n ( BH ) which maps to β .For a group L let ko + n ( BL ) be the subgroup of ko n ( BL ) given by D [ M, f ] where M is a positivelycurved spin manifold and f is a continuous map. In [3] the authors prove for any finite cyclic group H that ko + n ( BH ) = Ker( A ◦ p : ko n ( BH ) → KO n ( C ∗ r ( H ; R )). Thus, we have γ ∈ ko + n ( BH ) and β ∈ ko + n ( B Γ).Now, in [33] it is proven that if D [ M, f ] ∈ ko + n ( BG ), then M admits a metric of positive scalar curvature.In particular, we are done. 14n alternative direct proof of the calculations of the K -groups of PSL ( Z [ p ]) when p ≡
11 (mod 12)is given as follows. Note that this bypasses the computation of the Bredon homology but does not giveus a means to compute either invariant for SL ( Z [ p ]). Alternative proof of Theorem 1.5.
Let Λ by a set of representatives of finite subgroups of Γ. By [6,Theorem 4.1(a)] we have a short exact sequence0 → M ( H ) ∈ Λ e K n ( C ∗ r ( H )) → K n ( C ∗ r (Γ)) → K n (EΓ / Γ) → . The only nontrivial part now is computing K n (EΓ / Γ), but we have already shown that EΓ / Γ is homotopyequivalent to a wedge of W b (Γ ( p )) S , i.e. a wedge of 2-spheres. Thus, we can simply apply the homologicalAtiyah-Hirzebruch spectral sequence (which collapses trivially) to obtain that K (EΓ / Γ) = Z b (Γ ( p )+1) and K (EΓ / Γ) = 0.
Let Γ be a discrete group. In light of the proof of Theorem 1.7 we introduce the following condition onthe homotopy type of the suspension of EΓ / Γ.(WOS) n The space Σ(EΓ / Γ) has the homotopy type of a wedge of spheres of dimension less than n .We also introduce the following notation:(BC) Γ satisfies the Baum-Connes conjecture.(GLR) Γ satisfies the unstable GLR conjecture.(GLR) n Γ the unstable GLR conjecture for all manifolds of dimension greater than or equal to n . Corollary 6.3.
Let Γ be a discrete group satisfying (BC), (M), (NM) and (WOS) n , then Γ satisfies(GLR) n . In particular, if n ≤ then Γ satisfies (GLR). Let X = EΓ / Γ. One source of examples of groups satisfying (WOC) n +2 , for some n , is provided bygroups Γ such that H ∗ ( X ; Z ) is torsion-free and concentrated in at most 2 consecutive non-zero dimensions(dimensions n and n + 1). In this case we have a cofibration sequence: ∨ k S n → ∨ l S n → X → ∨ k S n +1 → ∨ l S n +1 → Σ X → ∨ k S n +2 → · · · Or if H ∗ ( X ; Z ) is concentrated in one non-zero degree: ∨ k S n → ∗ → X → ∨ k S n +1 → ∗ → Σ X → ∨ k S n +2 → · · · Now, applying H ∗ to either sequence, from the Hurewicz Theorem we obtain a homotopy equivalenceΣ X ≃ ( ∨ k S n +2 ) ∨ ( ∨ l S n +1 ) or Σ X ≃ ∨ k S n +2 .In particular, arbitrary free products of the following groups satisfy (GLR) n : One relator groups wherethe exponent sum of each letter in the relation is equal to 0, 1 or −
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