aa r X i v : . [ m a t h . K T ] J a n AN INTRODUCTION TO TORSION SUBCOMPLEX REDUCTION
ALEXANDER D. RAHM
Abstract.
This survey paper introduces to a technique called Torsion Subcomplex Reduc-tion (TSR) for computing torsion in the cohomology of discrete groups acting on suitable cellcomplexes. TSR enables one to skip machine computations on cell complexes, and to accessdirectly the reduced torsion subcomplexes, which yields results on the cohomology of matrixgroups in terms of formulas. TSR has already yielded general formulas for the cohomology ofthe tetrahedral Coxeter groups as well as, at odd torsion, of SL groups over arbitrary numberrings. The latter formulas have allowed to refine the Quillen conjecture. Furthermore, progresshas been made to adapt TSR to Bredon homology computations. In particular for the Bianchigroups, yielding their equivariant K -homology, and, by the Baum–Connes assembly map, the K -theory of their reduced C ∗ -algebras. As a side application, TSR has allowed to provide di-mension formulas for the Chen–Ruan orbifold cohomology of the complexified Bianchi orbifolds,and to prove Ruan’s crepant resolution conjecture for all complexified Bianchi orbifolds. Introduction
This survey paper is based on the habilitation thesis of the author, restricting to the ex-pository parts, which are updated here, and referring to previously published papers for theproofs. The goal is to introduce to a technique for computing Farrell cohomology of arithmeticgroups, presented in Section 3. This technique can also be applied in the computation of otherinvariants, as described in Section 2, where further results are stated.1.1.
Background.
Our objects of study are discrete groups Γ such that Γ admits a torsion-freesubgroup of finite index. By a theorem of Serre [39], all the torsion-free subgroups of finite indexin Γ have the same cohomological dimension; this dimension is called the virtual cohomologicaldimension (abbreviated vcd) of Γ. Above the vcd, the (co)homology of a discrete group isdetermined by its system of finite subgroups. We are going to discuss it in terms of Farrell–Tatecohomology (which we will by now just call Farrell cohomology). The Farrell cohomology b H q isidentical to group cohomology H q in all degrees q above the vcd, and extends in lower degreesto a cohomology theory of the system of finite subgroups. Details are elaborated in Brown’sbook [7, chapter X]. So for instance considering the Coxeter groups, the virtual cohomologicaldimension of all of which vanishes, their Farrell cohomology is identical to all of their groupcohomology. In Section 3.1, we will introduce a method of how to explicitly determine theFarrell cohomology: By reducing torsion sub-complexes.Let us note that for the same arithmetic groups, cohomology outside of our setting has muchstronger contemporary interest, and therefore, there has been extensive work on it. Just tomention a few, fairly recent publications about group cohomology in low cohomological degrees,by the editors of this special issue, from which to find more references: By Elbaz-Vincent et al.[13] for SL N ( Z ) with rising rank Z and modulo small torsion, by Ellis et al. [15] for arbitrary Date : 19th January 2021.2010
Mathematics Subject Classification.
MSC 11F75: Cohomology of arithmetic groups. roups using general purpose algorithms, by Gunnells et al. and S¸eng¨un et al. [3, 4] for infinitetowers of congruence subgroups.1.2. Overview of the results.
This paper shall introduce to the technique of torsion sub-complex reduction . It is a technique for the study of discrete groups Γ, giving easier access tothe cohomology of the latter at a fixed prime ℓ and above the virtual cohomological dimension,by extracting the relevant portion of the equivariant spectral sequence and then simplifying it.Instead of having to work with a full cellular complex X with a nice Γ-action, the techniqueinputs only an often lower-dimensional subcomplex of X , and reduces it to a small number ofcells.The author first developed torsion subcomplex reduction for a specific class of arithmeticgroups, the Bianchi groups, for which the method yielded all of the homology above the virtualcohomological dimension [30]. Some elements of this technique had already been used by Soul´efor a modular group [40]; and were used by Mislin and Henn as a set of ad hoc tricks. Afterrediscovering these ad hoc tricks, the author did put them into a general framework [29]. Theadvantage of using a systematic technique rather than a set of ad-hoc tricks is that instead ofmerely allowing for isolated ad-hoc example calculations, it becomes possible to find generalformulas, for instance for the entire family of the Bianchi groups.It is convenient to give some examples of where the technique of torsion subcomplex reductionhas already produced good results: • The Bianchi groups and their congruence subgroups (cf. Section 2.1); • The Coxeter groups (cf. Section 2.2); • The SL groups over arbitrary number rings (cf. Section 2.3); • PSL ( Z ) and the GL groups over rings of quadratic integers (cf. Section 2.4).This has led to the following applications: • Refining the Quillen conjecture (cf. Section 2.5), • Computing equivariant K -homology (cf. Section 2.6), • Understanding Chen–Ruan orbifold cohomology (cf. Section 2.7).The technique has also been adapted to groups with non-trivial centre (cf. Section 2.8).2.
Results
This section is going to state the results. Then in Section 3, we will have a more detailedlook at the methods.2.1.
The Bianchi groups and their congruence subgroups.
In the case of the PSL groupsover rings of imaginary quadratic integers (known as the Bianchi groups), the torsion subcom-plex reduction technique has permitted the author to find a description of the cohomology ringof these groups in terms of elementary number-theoretic quantities [29]. The key step has beento extract, using torsion subcomplex reduction, the essential information about the geometricmodels, and then to detach the cohomological information completely from the model.Torsion subcomplex reduction has then been combined with an analysis of the equivariantspectral sequence by Ethan Berkove, Grant Lakeland and the author to provide new tools for thecalculation of the torsion in the cohomology of congruence subgroups in the Bianchi groups [6]. .2. The Coxeter groups.
Let us recall that the Coxeter groups are generated by reflections,and their homology consists solely of torsion. Thus, torsion subcomplex reduction allows one toobtain all homology groups for all of the tetrahedral Coxeter groups at all odd prime numbers,in terms of a general formula [29].2.3.
The SL groups over arbitrary number rings. Matthias Wendt and the author haveestablished a complete description of the Farrell–Tate cohomology with odd torsion coefficientsfor all groups SL ( O K,S ), where O K,S is the ring of S-integers in an arbitrary number field K at an arbitrary non-empty finite set S of places of K containing the infinite places [35], basedon an explicit description of conjugacy classes of finite cyclic subgroups and their normalizersin SL ( O K,S ).The statement uses the following notation. Let ℓ be an odd prime number different from thecharacteristic of K . In the situation where, for ζ ℓ some primitive ℓ -th root of unity, ζ ℓ + ζ − ℓ ∈ K ,we will abuse notation and write O K,S [ ζ ℓ ] to mean the ring O K,S [ T ] / ( T − ( ζ ℓ + ζ − ℓ ) T + 1).Moreover, we denote the norm maps for class groups and units byNm : f K ( O K,S [ ζ ℓ ]) → f K ( O K,S ) and Nm : O K,S [ ζ ℓ ] × → O × K,S . Denote by M ( ℓ ) the ℓ -primary part of a module M ; by N G (Γ) the normalizer of Γ in G ; and by b H • Farrell cohomology (cf. Section 1.1).
Theorem 1 ([35]) . (1) b H • (SL ( O K,S ) , F ℓ ) = 0 if and only if ζ ℓ + ζ − ℓ ∈ K and the Steinitz class det O K,S ( O K,S [ ζ ℓ ]) is contained in the image of thenorm map Nm .(2) Assume the condition in (1) is satisfied. The set C ℓ of conjugacy classes of order ℓ elements in SL ( O K,S ) sits in an extension → coker Nm → C ℓ → ker Nm → . The set K ℓ of conjugacy classes of order ℓ subgroups of SL ( O K,S ) can be identified withthe quotient K ℓ = C ℓ / Gal( K ( ζ ℓ ) /K ) . There is a direct sum decomposition b H • (SL ( O K,S ) , F ℓ ) ∼ = M [Γ] ∈K ℓ b H • ( N SL ( O K,S ) (Γ) , F ℓ ) which is compatible with the ring structure, i.e., the Farrell-Tate cohomology ring of SL ( O K,S ) is a direct sum of the sub-rings for the normalizers N SL ( O K,S ) (Γ) .(3) If the class of Γ is not Gal( K ( ζ ℓ ) /K ) -invariant, then N SL ( O K,S ) (Γ) ∼ = ker Nm . There is a degree cohomology class a and a ring isomorphism b H • (ker Nm , Z ) ( ℓ ) ∼ = F ℓ [ a , a − ] ⊗ F ℓ ^ (ker Nm ) . In particular, this is a free module over the subring F ℓ [ a , a − ] .(4) If the class of Γ is Gal( K ( ζ ℓ ) /K ) -invariant, then there is an extension → ker Nm → N SL ( O K,S ) (Γ) → Z / → . here is a ring isomorphism b H • ( N SL ( O K,S ) (Γ) , Z ) ( ℓ ) ∼ = (cid:16) F ℓ [ a , a − ] ⊗ F ℓ ^ (ker Nm ) (cid:17) Z / , with the Z / -action given by multiplication with − on a and ker Nm . In particular,this is a free module over the subring F ℓ [ a , a − ] ∼ = b H • ( D ℓ , Z ) ( ℓ ) . (5) The restriction map induced from the inclusion SL ( O K,S ) → SL ( C ) maps the secondChern class c to the sum of the elements a in all the components. Wendt has furthermore extended this investigation to the cases of SL over the ring of func-tions on a smooth affine curve over an algebraically closed field [43].2.4. Farrell–Tate cohomology of higher rank arithmetic groups.
Pertinent progresswas also made on the Farrell–Tate cohomology of GL over rings of quadratic integers. Forthis purpose, the conjugacy classification of cyclic subgroups was reduced to the classificationof modules of group rings over suitable rings of integers which are principal ideal domains,generalizing an old result of Reiner. As an example of the number-theoretic input requiredfor the Farrell-Tate cohomology computations, Bui, Wendt and the author did describe thehomological torsion in PGL over principal ideal rings of quadratic integers, accompanied bymachine computations in the imaginary quadratic case [10].For machine calculations of Farrell–Tate or Bredon (co)homology, one needs cell complexeswhere cell stabilizers fix their cells pointwise. Bui, Wendt and the author have provided twoalgorithms computing an efficient subdivision of a complex to achieve this rigidity property [33].Applying these algorithms to available cell complexes for PSL ( Z ), they have computed theFarrell–Tate cohomology for small primes as well as the Bredon homology for the classifyingspaces of proper actions with coefficients in the complex representation ring.2.5. Investigation of the refined Quillen conjecture.
The Quillen conjecture on the co-homology of arithmetic groups has spurred a great deal of mathematics (see the pertinentmonograph [20]). Using Farrell–Tate cohomology computations, Wendt and the author haveestablished further positive cases for the Quillen conjecture for SL . In detail, the originalconjecture of 1971 [26] is as follows for GL n . Conjecture 2 (Quillen) . Let ℓ be a prime number. Let K be a number field with ζ ℓ ∈ K ,and S a finite set of places containing the infinite places and the places over ℓ . Then thenatural inclusion O K,S ֒ → C makes H • (GL n ( O K,S ) , F ℓ ) a free module over the cohomology ring H • cts (GL n ( C ) , F ℓ ) . While there are counterexamples to the original version of the conjecture, it holds true inmany other cases. From the first counterexamples through the present, the conjecture has keptresearchers interested in determining its range of validity [2].Positive cases in which the conjecture has been established are n = ℓ = 2 by Mitchell [24], n = 3, ℓ = 2 by Henn [17], and n = 2, ℓ = 3 by Anton [1].On the other hand, cases where the Quillen conjecture is known to be false can all be tracedto a remark by Henn, Lannes and Schwartz [18, remark on p. 51], which shows that Quillen’s onjecture for GL n ( Z [1 / • (GL n ( Z [1 / , F ) → H • (T n ( Z [1 / , F )from GL n ( Z [1 / n ( Z [1 / n ≥
32 and ℓ = 2. Dwyer’s bound wassubsequently improved by Henn and Lannes to n ≥
14. At the prime ℓ = 3, Anton [1] provednon-injectivity for n ≥ to statements about Steinberg homology. This, together with the results of [43], hasallowed us to find a refined version of the Quillen conjecture, which keeps track of all the typesof known counter-examples to the original Quillen conjecture: Conjecture 3 (Refined Quillen conjecture [36]) . Let K be a number field. Fix a prime ℓ suchthat ζ ℓ ∈ K , and an integer n < ℓ . Assume that S is a set of places containing the infiniteplaces and the places lying over ℓ . If each cohomology class of GL n ( O K,S ) is detected on somefinite subgroup, then H • (GL n ( O K,S ) , F ℓ ) is a free module over the image of the restriction map H • cts (GL n ( C ) , F ℓ ) → H • (GL n ( O K,S ) , F ℓ ) . We can make the following use of the description of the Farrell–Tate cohomology of SL overrings of S -integers. Corollary 4 (Corollary to Theorem 1) . Let K be a number field, let S be a finite set of placescontaining the infinite ones, and let ℓ be an odd prime.(1) The original Quillen conjecture holds for group cohomology H • (SL ( O K,S ) , F ℓ ) above thevirtual cohomological dimension.(2) The refined Quillen conjecture holds for Farrell–Tate cohomology b H • (SL ( O K,S ) , F ℓ ) . Verification of the Quillen conjecture in the rank 2 imaginary quadratic case.
Buiand the author did confirm a conjecture of Quillen in the case of the mod 2 cohomology ofarithmetic groups SL ( O Q ( √− m ) [ ]), where O Q ( √− m ) is an imaginary quadratic ring of integers.To make explicit the free module structure on the cohomology ring conjectured by Quillen, theycomputed the mod 2 cohomology of SL ( Z [ √− ]) via the amalgamated decomposition of thelatter group [8].2.6. Application to equivariant K -homology. For the Bianchi groups, the torsion subcom-plex reduction technique was adapted from group homology to Bredon homology H
Fin n (Γ; R C )with coefficients in the complex representation rings, and with respect to the family of finitesubgroups [31]. This has led the author to the following formulas for this Bredon homology, andby the Atiyah–Hirzebruch spectral sequence, to the below formulas for equivariant K -homologyof the Bianchi groups acting on their classifying space for proper actions. Theorem 5.
Let Γ be a Bianchi group or any one of its subgroups. Then the Bredon homology H Fin n (Γ; R C ) splits as a direct sum over (1) the orbit space homology H n (BΓ; Z ) ,(2) a submodule H n (Ψ (2) • ) determined by the reduced -torsion subcomplex of (EΓ , Γ) and(3) a submodule H n (Ψ (3) • ) determined by the reduced -torsion subcomplex of (EΓ , Γ) . hese submodules are given as follows.Except for the Gaussian and Eisenstein integers, which can easily be treated ad hoc [27], allthe rings of integers of imaginary quadratic number fields admit as only units {± } . In thelatter case, we call PSL ( O − m ) a Bianchi group with units {± } . Theorem 6.
The -torsion part of the Bredon complex of a Bianchi group Γ with units {± } has homology H n (Ψ (2) • ) ∼ = Z z ⊕ ( Z / d , n = 0 , Z o , n = 1 , , otherwise , where z counts the number of conjugacy classes of subgroups of type Z / in Γ , o counts theconjugacy classes of type Z / in Γ which are not contained in any -dihedral subgroup, and d counts the number of -dihedral subgroups, whether or not they are contained in a tetrahedralsubgroup of Γ . Theorem 7.
The -torsion part of the Bredon complex of a Bianchi group Γ with units {± } has homology H n (Ψ (3) • ) ∼ = Z o + ι , n = 0 or 1 , , otherwise , where amongst the subgroups of type Z / in Γ , o counts the number of conjugacy classes ofthose of them which are not contained in any -dihedral subgroup, and ι counts the conjugacyclasses of those of them which are contained in some -dihedral subgroup in Γ . There are formulas for o , z , d , o and ι in terms of elementary number-theoretic quant-ities [21], which are readily computable by machine [29, appendix]. See Table 2 for how theyrelate to the types of connected components of torsion subcomplexes.We deduce the following formulas for the equivariant K -homology of the Bianchi groups.Note for this purpose that for a Bianchi group Γ, there is a model for EΓ of dimension 2, soH (BΓ; Z ) ∼ = Z β is torsion-free. Note also that the naive Euler characteristic of the Bianchigroups vanishes (again excluding the two special cases of Gaussian and Eisensteinian integers),that is, for β i = dim H i (BΓ; Q ) we have β − β + β = 0 and β = 1. Corollary 8.
For any Bianchi group Γ with units {± } , the short exact sequence linking Bredonhomology and equivariant K -homology splits into K Γ0 (EΓ) ∼ = Z ⊕ Z β ⊕ Z z ⊕ ( Z / d ⊕ Z o + ι . Furthermore, K Γ1 (EΓ) ∼ = H (BΓ; Z ) ⊕ Z o ⊕ Z o + ι . .7. Chen–Ruan orbifold cohomology of the complexified Bianchi orbifolds.
The ac-tion of the Bianchi groups Γ on real hyperbolic 3-space (SL ( C ) / SU ) induces an action on acomplexification (SL ( C ) / SU ) C of the latter (of real dimension 6).For the orbifolds [(SL ( C ) / SU ) C / Γ] given by this action, we can compute the Chen–RuanOrbifold Cohomology as follows.
Theorem 9 ([25]) . Let Γ be a finite index subgroup in a Bianchi group (except over the Gaussianor Eisensteinian integers). Denote by λ n the number of conjugacy classes of cyclic subgroupsof order n in Γ . Denote by λ ∗ n the cardinality of the subset of conjugacy classes which arecontained in a dihedral subgroup of order n in Γ . Then, H dorb ([(SL ( C ) / SU ) C / Γ]) ∼ = H d ((SL ( C ) / SU ) / Γ ; Q ) ⊕ Q λ +2 λ − λ ∗ , d = 2 , Q λ − λ ∗ +2 λ − λ ∗ , d = 3 , , otherwise . The (co)homology of the quotient space (SL ( C ) / SU ) / Γ has been computed numerically fora large range of Bianchi groups [41], [37], [32]; and bounds for its Betti numbers have been givenin [22]. Kr¨amer [21] has determined number-theoretic formulas for the numbers λ n and λ ∗ n ofconjugacy classes of finite subgroups in the Bianchi groups.Building on this, Perroni and the author have established the following result [25]. Theorem 10.
Let (SL ( C ) / SU ) C / Γ be the coarse moduli space of [(SL ( C ) / SU ) C / Γ] ;and let Y → (SL ( C ) / SU ) C / Γ be a crepant resolution of (SL ( C ) / SU ) C / Γ .Then there is an isomorphism as graded C -algebras between the Chen-Ruan cohomology ring of [(SL ( C ) / SU ) C / Γ] and the singular cohomology ring of Y : ( H ∗ CR ([(SL ( C ) / SU ) C / Γ]) , ∪ CR ) ∼ = ( H ∗ ( Y ) , ∪ ) . The Chen–Ruan orbifold cohomology is conjectured by Ruan to match the quantum correctedclassical cohomology ring of a crepant resolution for the orbifold. Perroni and the author haveproved furthermore that the Gromov-Witten invariants involved in the definition of the quantumcorrected cohomology ring of Y → (SL ( C ) / SU ) C / Γ vanish. Hence, they have deduced thefollowing. Corollary 11.
Ruan’s crepant resolution conjecture holds true for the complexified Bianchiorbifolds [(SL ( C ) / SU ) C / Γ] . Adaptation of the technique to groups with non-trivial centre.
Berkove and theauthor [5] have extended the technique of torsion subcomplex reduction, which originally wasdesigned for groups with trivial centre (e.g., PSL ), to groups with non-trivial centre (e.g.,SL ). This way, they have determined the 2-torsion in the cohomology of the SL groups overimaginary quadratic number rings O − m in Q ( √− m ), based on their action on hyperbolic 3-space H . For instance, they get the following result in the case where the quotient of the2–torsion subcomplex has the shape b b , which is equivalent to the following three conditions(cf. [29]): m ≡ Q ( √− m ) has precisely one finite ramification place over Q , andthe ideal class number of the totally real number field Q ( √ m ) is 1. Under these assumptions, ur cohomology ring has the following dimensions:dim F H q (SL ( O − m ) ; F ) = β + β , q = 4 k + 5 ,β + β + 2 , q = 4 k + 4 ,β + β + 3 , q = 4 k + 3 ,β + β + 1 , q = 4 k + 2 ,β , q = 1 , where β q := dim F H q ( SL ( O − m ) \H ; F ). Let β := dim Q H ( SL ( O − m ) \H ; Q ) . For all absolutevalues of the discriminant less than 296, numerical calculations yield β + 1 = β = β . In thisrange, the numbers m subject to the above dimension formula and β are given as follows (theBetti numbers have been computed in a previous paper of the author [32]). m
11 19 43 59 67 83 107 131 139 163 179 211 227 251 283 β The technique of Torsion Subcomplex Reduction
Reduction of torsion subcomplexes in the classical setting.
Let ℓ be a prime num-ber. We require any discrete group Γ under our study to be provided with what we will calla polytopal Γ -cell complex , that is, a finite-dimensional simplicial complex X with cellular Γ-action such that each cell stabiliser fixes its cell point-wise. In practice, we relax the simplicialcondition to a polyhedral one, merging finitely many simplices to a suitable polytope. We couldobtain the simplicial complex back as a triangulation. We further require that the fixed pointset X G be acyclic for every non-trivial finite ℓ -subgroup G of Γ.Then, the Γ-equivariant Farrell cohomology b H ∗ Γ ( X ; M ) of X , for any trivial Γ-module M ofcoefficients, gives us the ℓ -primary part b H ∗ (Γ; M ) ( ℓ ) of the Farrell cohomology of Γ, as follows. Proposition 12 (Brown [7]) . For a Γ -action on X as specified above, the canonical map b H ∗ (Γ; M ) ( ℓ ) → b H ∗ Γ ( X ; M ) ( ℓ ) is an isomorphism. The classical choice [7] is to take for X the geometric realization of the partially ordered set ofnon-trivial finite subgroups (respectively, non-trivial elementary Abelian ℓ -subgroups) of Γ, thelatter acting by conjugation. The stabilisers are then the normalizers, which in many discretegroups are infinite. In addition, there are often great computational challenges to determine agroup presentation for the normalizers. When we want to compute the module b H ∗ Γ ( X ; M ) ( ℓ ) subject to Proposition 12, at least we must know the ( ℓ -primary part of the) Farrell cohomologyof these normalizers. The Bianchi groups are an instance where different isomorphism typescan occur for this cohomology at different conjugacy classes of elementary Abelian ℓ -subgroups,both for ℓ = 2 and ℓ = 3. As the only non-trivial elementary Abelian 3-subgroups in the Bianchigroups are of rank 1, the orbit space Γ \ X consists only of one point for each conjugacy class oftype Z / o the different possible homological types of the normalizers (in fact, two of them occur), thefinal result remains unclear and subject to tedious case-by-case computations of the normalizers.In contrast, in the cell complex we are going to construct (specified in Definition 16 below), theconnected components of the orbit space are for the 3-torsion in the Bianchi groups not simplepoints, but have either the shape b b or b . This dichotomy already contains the informationabout the occurring normalizer.The starting point for our construction is the following definition. Definition 13.
Let ℓ be a prime number. The ℓ -torsion subcomplex of a polytopal Γ-cellcomplex X consists of all the cells of X whose stabilisers in Γ contain elements of order ℓ .We are from now on going to require the cell complex X to admit only finite stabilisers in Γ,and we require the action of Γ on the coefficient module M to be trivial. Then obviously onlycells from the ℓ -torsion subcomplex contribute to b H ∗ Γ ( X ; M ) ( ℓ ) . Corollary 14 (Corollary to Proposition 12) . There is an isomorphism between the ℓ -primaryparts of the Farrell cohomology of Γ and the Γ -equivariant Farrell cohomology of the ℓ -torsionsubcomplex. We are going to reduce the ℓ -torsion subcomplex to one which still carries the Γ-equivariantFarrell cohomology of X , but which can also have considerably fewer orbits of cells. This can beeasier to handle in practice, and, for certain classes of groups, leads us to an explicit structuraldescription of the Farrell cohomology of Γ. The pivotal property of this reduced ℓ -torsionsubcomplex will be given in Theorem 17. Our reduction process uses the following conditions,which are imposed to a triple ( σ, τ , τ ) of cells in the ℓ -torsion subcomplex, where σ is a cellof dimension n −
1, lying in the boundary of precisely the two n -cells τ and τ , the latter cellsrepresenting two different orbits. Condition A.
The triple ( σ, τ , τ ) is said to satisfy Condition A if no higher-dimensionalcells of the ℓ -torsion subcomplex touch σ ; and if the n -cell stabilisers admit an isomorphismΓ τ ∼ = Γ τ .Where this condition is fulfilled in the ℓ -torsion subcomplex, we merge the cells τ and τ along σ and do so for their entire orbits, if and only if they meet the following additionalcondition, that we never merge two cells the interior of which contains two points on the sameorbit. We will refer by mod ℓ cohomology to group cohomology with Z /ℓ -coefficients under thetrivial action. Condition B.
With the notation above Condition A , the inclusion Γ τ ⊂ Γ σ induces anisomorphism on mod ℓ cohomology. Lemma 15 ([29]) . Let g X ( ℓ ) be the Γ -complex obtained by orbit-wise merging two n -cells of the ℓ -torsion subcomplex X ( ℓ ) which satisfy Conditions A and B . Then, b H ∗ Γ ( g X ( ℓ ) ; M ) ( ℓ ) ∼ = b H ∗ Γ ( X ( ℓ ) ; M ) ( ℓ ) . By a “terminal ( n − n − σ with • modulo Γ precisely one adjacent n -cell τ , • and such that τ has no further cells on the Γ-orbit of σ in its boundary; furthermore there shall be no higher-dimensional cells adjacent to σ .And by “cutting off” the n -cell τ , we will mean that we remove τ together with σ from our cellcomplex. Definition 16. A reduced ℓ -torsion subcomplex associated to a polytopal Γ-cell complex X is a cell complex obtained by recursively merging orbit-wise all the pairs of cells satisfyingconditions A and B , and cutting off n -cells that admit a terminal ( n − B is satisfied.A priori, this process yields a unique reduced ℓ -torsion subcomplex only up to suitable iso-morphisms, so we do not speak of “the” reduced ℓ -torsion subcomplex. The following theoremmakes sure that the Γ-equivariant mod ℓ Farrell cohomology is not affected by this issue.
Theorem 17 ([29]) . There is an isomorphism between the ℓ -primary part of the Farrell cohomo-logy of Γ and the Γ -equivariant Farrell cohomology of a reduced ℓ -torsion subcomplex obtainedfrom X as specified above. In order to have a practical criterion for checking Condition B , we make use of the followingstronger condition.Here, we write N Γ σ for taking the normalizer in Γ σ and Sylow ℓ for picking an arbitrarySylow ℓ -subgroup. This is well defined because all Sylow ℓ -subgroups are conjugate. We useZassenhaus’s notion for a finite group to be ℓ - normal , if the center of one of its Sylow ℓ -subgroupsis the center of every Sylow ℓ -subgroup in which it is contained. Condition B’.
With the notation of Condition A , the group Γ σ admits a (possibly trivial)normal subgroup T σ with trivial mod ℓ cohomology and with quotient group G σ ; and the groupΓ τ admits a (possibly trivial) normal subgroup T τ with trivial mod ℓ cohomology and withquotient group G τ making the sequences1 → T σ → Γ σ → G σ → → T τ → Γ τ → G τ → G τ ∼ = G σ , or(2) G σ is ℓ -normal and G τ ∼ = N G σ (center(Sylow ℓ ( G σ ))), or(3) both G σ and G τ are ℓ -normal and there is a (possibly trivial) group T with trivial mod ℓ cohomology making the sequence1 → T → N G σ (center(Sylow ℓ ( G σ ))) → N G τ (center(Sylow ℓ ( G τ ))) → Lemma 18 ([29]) . Condition B’ implies Condition B.
Remark 19.
The computer implementation [9] checks Conditions B ′ (1) and B ′ (2) for eachpair of cell stabilisers, using a presentation of the latter in terms of matrices, permutationcycles or generators and relators. In the below examples however, we do avoid this case-by-casecomputation by a general determination of the isomorphism types of pairs of cell stabilisers forwhich group inclusion induces an isomorphism on mod ℓ cohomology. The latter method is theprocedure of preference, because it allows us to deduce statements that hold for the entire classof groups in question. .1.1. Example: A -torsion subcomplex for SL ( Z ) . The 2-torsion subcomplex of the cell com-plex described by Soul´e [40], obtained from the action of SL ( Z ) on its symmetric space, hasthe following homeomorphic image. stab(M) ∼ = S stab(Q) ∼ = D stab(O) ∼ = S stab(N) ∼ = D stab(P) ∼ = S N’ M’ D D D D Z / Z / Z / D Z / D D Z / Here, the three edges
N M , N M ′ and N ′ M ′ have to be identified as indicated by the arrows. Allof the seven triangles belong with their interior to the 2-torsion subcomplex, each with stabiliser Z /
2, except for the one which is marked to have stabiliser D . Using the methods described inSection 3.1, we reduce this subcomplex to b S O b D D Q Z / S M b D S P b D D N ′ b and then to S b Z / S b D S b which is the geometric realization of Soul´e’s diagram of cell stabilisers. This yields the mod 2Farrell cohomology as specified in [40].3.1.2. Example: Farrell cohomology of the Bianchi modular groups.
Consider the SL matrixgroups over the ring O − m of integers in the imaginary quadratic number field Q ( √− m ), with m a square-free positive integer. These groups, as well as their central quotients PSL ( O − m ),are known as Bianchi (modular) groups . We recall the following information from [29] on the ℓ -torsion subcomplex of PSL ( O − m ). Let Γ be a finite index subgroup in PSL ( O − m ). Then ubgroup type Z / Z / D D A Number of conjugacy classes λ λ µ µ µ T Table 1.
The non-trivial finite subgroups of PSL ( O − m ) have been classifiedby Klein [19]. Here, Z /n is the cyclic group of order n , the dihedral groups are D with four elements and D with six elements, and the tetrahedral group isisomorphic to the alternating group A on four letters. Formulas for the numbersof conjugacy classes counted by the Greek symbols have been given byKr¨amer [21].any element of Γ fixing a point inside hyperbolic 3-space H acts as a rotation of finite order.By Felix Klein’s work, we know conversely that any torsion element α is elliptic and hencefixes some geodesic line. We call this line the rotation axis of α . Every torsion element acts asthe stabiliser of a line conjugate to one passing through the Bianchi fundamental polyhedron.We obtain the refined cellular complex from the action of Γ on H as described in [30], namelywe subdivide H until the stabiliser in Γ of any cell σ fixes σ point-wise. We achieve this bycomputing Bianchi’s fundamental polyhedron for the action of Γ, taking as a preliminary set of2-cells its facets lying on the Euclidean hemispheres and vertical planes of the upper-half spacemodel for H , and then subdividing along the rotation axes of the elements of Γ.It is well-known [38] that if γ is an element of finite order n in a Bianchi group, then n mustbe 1, 2, 3, 4 or 6, because γ has eigenvalues ρ and ρ , with ρ a primitive n -th root of unity, andthe trace of γ is ρ + ρ ∈ O − m ∩ R = Z . When ℓ is one of the two occurring prime numbers2 and 3, the orbit space of this subcomplex is a graph, because the cells of dimension greaterthan 1 are trivially stabilized in the refined cellular complex. We can see that this graph is finiteeither from the finiteness of the Bianchi fundamental polyhedron, or from studying conjugacyclasses of finite subgroups as in [21].As in [34], we make use of a 2-dimensional deformation retract X of the refined cellularcomplex, equivariant with respect to a Bianchi group Γ. This retract has a cell structure inwhich each cell stabiliser fixes its cell pointwise. Since X is a deformation retract of H andhence acyclic, H ∗ Γ ( X ) ∼ = H ∗ Γ ( H ) ∼ = H ∗ (Γ) . In Theorem 20 below, we give a formula expressing precisely how the Farrell cohomology ofa Bianchi group with units {± } (i.e., just excluding the Gaussian and the Eisentein integersas imaginary quadratic rings, see Section 2.6) depends on the numbers of conjugacy classesof non-trivial finite subgroups of the occurring five types specified in Table 1. The main stepin order to prove this, is to read off the Farrell cohomology from the quotient of the reducedtorsion sub-complexes.Kr¨amer’s formulas [21] express the numbers of conjugacy classes of the five types of non-trivial finite subgroups given in Table 1. We are going to use the symbols of that table also forthe numbers of conjugacy classes in Γ, where Γ is a finite index subgroup in a Bianchi group.Recall that for ℓ = 2 and ℓ = 3, we can express the the dimensions of the homology of Γ withcoefficients in the field F ℓ with ℓ elements in degrees above the virtual cohomological dimension f the Bianchi groups – which is 2 – by the Poincar´e series P ℓ Γ ( t ) := ∞ X q > dim F ℓ H q (Γ; F ℓ ) t q , which has been suggested by Grunewald. Further let P b ( t ) := − t t − , which equals the Poincar´eseries P ( t ) of the groups Γ the quotient of the reduced 2–torsion sub-complex of which is acircle. Denote by • P ∗D ( t ) := − t (3 t − t − , the Poincar´e series overdim F H q ( D ; F ) −
32 dim F H q ( Z / F ) • and by P ∗A ( t ) := − t ( t − t +2 t − t − ( t + t +1) , the Poincar´e series overdim F H q ( A ; F ) −
12 dim F H q ( Z / F ) . In 3-torsion, let P b b ( t ) := − t ( t − t +2)( t − t +1) , which equals the Poincar´e series P ( t ) for thoseBianchi groups, the quotient of the reduced 3–torsion sub-complex of which is a single edgewithout identifications. Theorem 20 ([29]) . For any finite index subgroup Γ in a Bianchi group with units {± } , thegroup homology in degrees above its virtual cohomological dimension is given by the Poincar´eseries P ( t ) = (cid:18) λ − µ − µ T (cid:19) P b ( t ) + ( µ − µ T ) P ∗D ( t ) + µ T P ∗A ( t ) and P ( t ) = (cid:16) λ − µ (cid:17) P b ( t ) + µ P b b ( t ) . More general results are stated in Section 2.3 above.3.1.3.
Example: Farrell cohomology of Coxeter (tetrahedral) groups.
Recall that a Coxeter groupis a group admitting a presentation h g , g , ..., g n | ( g i g j ) m i,j = 1 i , where m i,i = 1; for i = j we have m i,j ≥
2; and m i,j = ∞ is permitted, meaning that ( g i g j )is not of finite order. As the Coxeter groups admit a contractible classifying space for properactions [12], their Farrell cohomology yields all of their group cohomology. So in this section,we make use of this fact to determine the latter. For facts about Coxeter groups, and especiallyfor the Davis complex, we refer to [12]. Recall that the simplest example of a Coxeter group,the dihedral group D n , is an extension1 → Z /n → D n → Z / → . So we can make use of the original application [42] of Wall’s lemma to obtain its mod ℓ homologyfor prime numbers ℓ > q ( D n ; Z /ℓ ) ∼ = Z /ℓ, q = 0 , Z / gcd( n, ℓ ) , q ≡ , , otherwise . heorem 21 ([29]) . Let ℓ > be a prime number. Let Γ be a Coxeter group admitting aCoxeter system with at most four generators, and relator orders not divisible by ℓ . Let Z ( ℓ ) bethe ℓ –torsion sub-complex of the Davis complex of Γ . If Z ( ℓ ) is at most one-dimensional andits orbit space contains no loop or bifurcation, then the mod ℓ homology of Γ is isomorphic to (H q ( D ℓ ; Z /ℓ )) m , with m the number of connected components of the orbit space of Z ( ℓ ) . The conditions of this theorem are for instance fulfilled by the Coxeter tetrahedral groups;the exponent m has been specified for each of them in the tables in [29]. In the easier case ofCoxeter triangle groups, we can sharpen the statement as follows. The non-spherical and henceinfinite Coxeter triangle groups are given by the presentation h a, b, c | a = b = c = ( ab ) p = ( bc ) q = ( ca ) r = 1 i , where 2 ≤ p, q, r ∈ N and p + q + r ≤ Proposition 22 ([29]) . For any prime number ℓ > , the mod ℓ homology of a Coxeter trianglegroup is given as the direct sum over the mod ℓ homology of the dihedral groups D p , D q and D r . The non-central torsion subcomplex.
In the case of a trivial kernel of the actionon the polytopal Γ-cell complex, torsion subcomplex reduction allows one to establish generalformulas for the Farrell cohomology of Γ [29]. In contrast, for instance the action of SL ( O − m )on hyperbolic 3-space has the 2-torsion group {± } in the kernel; since every cell stabilisercontains 2-torsion, the 2-torsion subcomplex then does not ease our calculation in any way.We can remedy this situation by considering the following object, on whose cells we impose asupplementary property. Definition 23.
The non-central ℓ -torsion subcomplex of a polytopal Γ-cell complex X consistsof all the cells of X whose stabilisers in Γ contain elements of order ℓ that are not in the centerof Γ.We note that this definition yields a correspondence between, on one side, the non-central ℓ -torsion subcomplex for a group action with kernel the center of the group, and on the other side,the ℓ -torsion subcomplex for its central quotient group. In [5], this correspondence has beenused in order to identify the non-central ℓ -torsion subcomplex for the action of SL ( O − m ) onhyperbolic 3-space as the ℓ -torsion subcomplex of PSL ( O − m ). However, incorporating the non-central condition for SL ( O − m ) introduces significant technical obstacles, which were addressedin that paper, establishing the following theorem for any finite index subgroup Γ in SL ( O − m ).Denote by X a Γ-equivariant retract of SL ( C ) / SU , by X s the 2-torsion subcomplex withrespect to PΓ (the “non-central” 2-torsion subcomplex for Γ), and by X ′ s the part of it withhigher 2-rank. Further, let v denote the number of conjugacy classes of subgroups of higher2-rank, and define sign( v ) := , v = 0 , , v > . For q ∈ { , } , denote the dimension dim F H q ( Γ \ X ; F ) by β q . able 2. Connected component types of reduced torsion subcomplex quotientsfor the PSL Bianchi groups. The exhaustiveness of this table has been estab-lished using theorems of Kr¨amer [5].2–torsionsubcomplexcomponents countedby 3–torsionsubcomplexcomponents countedby b Z / o = λ − λ ∗ b Z / o = λ − λ ∗ A b b A ι D b b D ι = λ ∗ D b b D θ D b b A ρ Theorem 24 ([5]) . The E page of the equivariant spectral sequence with F -coefficients asso-ciated to the action of Γ on X is concentrated in the columns n ∈ { , , } and has the followingform. q = 4 k + 3 E , ( X s ) E , ( X s ) ⊕ ( F ) a ( F ) a q = 4 k + 2 H ( X ′ s ) ⊕ ( F ) − sign( v ) ( F ) a H ( Γ \ X ) q = 4 k + 1 E , ( X s ) E , ( X s ) ⊕ ( F ) a ( F ) a q = 4 k F H ( Γ \ X ) H ( Γ \ X ) k ∈ N ∪ { } n = 0 n = 1 n = 2 where a = χ ( Γ \ X s ) − β ( Γ \ X ) + ca = β ( Γ \ X ) + ca = β ( Γ \ X ) + v − sign( v ) . In order to derive the example stated in Section 2.8 above, we combine the latter theoremwith the following determination (carried out in [5]) of the d -differentials on the four possible(cf. Table 2) connected component types b , b b , b b and b b of the reduced non-central2-torsion subcomplex for the full SL groups over the imaginary quadratic number rings. Lemma 25 ([5]) . The d differential in the equivariant spectral sequence associated to theaction of SL ( O − m ) on hyperbolic space is trivial on components of the non-central -torsionsubcomplex quotient • of type b in dimensions q ≡ if and only if it is trivial on these components indimensions q ≡ . • of type b b . • of types b b and b b in dimensions q ≡ . .3. Application to equivariant K -homology. In order to adapt torsion subcomplex re-duction to Bredon homology and prove Theorem 5, we need to perform a “representation ringsplitting”.
Representation ring splitting . The classification of Felix Klein [19] of the finite subgroupsin PSL ( O ) is recalled in Table 1. We further use the existence of geometric models for theBianchi groups in which all edge stabilisers are finite cyclic and all cells of dimension 2 andhigher are trivially stabilised. Therefore, the system of finite subgroups of the Bianchi groupsadmits inclusions only emanating from cyclic groups. This makes the Bianchi groups and theirsubgroups subject to the splitting of Bredon homology stated in Theorem 5.The proof of Theorem 5 is based on the above particularities of the Bianchi groups, andapplies the following splitting lemma for the involved representation rings to a Bredon complexfor (EΓ , Γ).
Lemma 26 ([31]) . Consider a group Γ such that every one of its finite subgroups is either cyclicof order at most , or of one of the types D , D or A . Then there exist bases of the complexrepresentation rings of the finite subgroups of Γ , such that simultaneously every morphism ofrepresentation rings induced by inclusion of cyclic groups into finite subgroups of Γ , splits as amatrix into the following diagonal blocks.(1) A block of rank induced by the trivial and regular representations,(2) a block induced by the –torsion subgroups(3) and a block induced by the –torsion subgroups. As this splitting holds simultaneously for every morphism of representation rings, we havesuch a splitting for every morphism of formal sums of representation rings, and hence for thedifferential maps of the Bredon complex for any Bianchi group and any of their subgroups.The bases that are mentioned in the above lemma, are obtained by elementary base trans-formations from the canonical basis of the complex representation ring of a finite group to abasis whose matrix form has • its first row concentrated in its first entry, for a finite cyclic group (edge stabiliser). Thebase transformation is carried out by summing over all representations to replace thetrivial representation by the regular representation. • its first column concentrated in its first entry, for a finite non-cyclic group (vertex sta-biliser). The base transformation is carried out by subtracting the trivial representationfrom each representation, except from itself.The details are provided in [31].In this setting, the technique has inspired work beyond the range of arithmetic groups, whichhas led to formulas for the integral Bredon homology and equivariant K-homology of all compact3-dimensional hyperbolic reflection groups [23], through a novel criterion for torsion-freeness ofequivariant K-homology in a more general framework.3.4. Chen–Ruan orbifold cohomology of the complexified orbifolds.
Let Γ be a discretegroup acting properly , i.e. with finite stabilizers, by diffeomorphisms on a manifold Y . For anyelement g ∈ Γ, denote by C Γ ( g ) the centralizer of g in Γ. Denote by Y g the subset of Y consisting of the fixed points of g . efinition 27. Let T ⊂ Γ be a set of representatives of the conjugacy classes of elements offinite order in Γ. Then we setH ∗ orb ([ Y /
Γ]) := M g ∈ T H ∗ ( Y g /C Γ ( g ); Q ) . It can be checked that this definition gives the vector space structure of the orbifold cohomo-logy defined by Chen and Ruan [11], if we forget the grading of the latter. We can verify thisfact using arguments analogous to those used by Fantechi and G¨ottsche [16] in the case of afinite group Γ acting on Y . The additional argument needed when considering some element g in Γ of infinite order, is the following. As the action of Γ on Y is proper, g does not admit anyfixed point in Y . Thus, H ∗ ( Y g /C Γ ( g ); Q ) = H ∗ ( ∅ ; Q ) = 0 . A result on the vector space structure of the Chen–Ruan orbifold cohomology of Bianchiorbifolds are the below two theorems.
Theorem 28 ([31]) . For any element γ of order in a finite index subgroup Γ in a Bianchigroup with units {± } , the quotient space H γ / C Γ ( γ ) of the rotation axis modulo the centralizerof γ is homeomorphic to a circle. Theorem 29 ([31]) . Let γ be an element of order in a Bianchi group Γ with units {± } .Then, the homeomorphism type of the quotient space H γ / C Γ ( γ ) is b b an edge without identifications, if h γ i is contained in a subgroup of type D inside Γ and b a circle, otherwise. Denote by λ ℓ the number of conjugacy classes of subgroups of type Z / ℓ Z in a finite indexsubgroup Γ in a Bianchi group with units {± } . Denote by λ ∗ ℓ the number of conjugacy classesof subgroups of type Z / ℓ Z which are contained in a subgroup of type D n in Γ. By [31], thereare 2 λ − λ ∗ conjugacy classes of elements of order 3. As a result of Theorems 28 and 29, thevector space structure of the orbifold cohomology of [ H R / Γ] is given asH • orb ([ H R / Γ]) ∼ = H • ( H R / Γ ; Q ) L λ ∗ H • (cid:16) b b ; Q (cid:17) L ( λ − λ ∗ ) H • (cid:0) b ; Q (cid:1) L (2 λ − λ ∗ ) H • (cid:0) b ; Q (cid:1) . The (co)homology of the quotient space H R / Γ has been computed numerically for a large rangeof Bianchi groups [41], [37], [32]; and bounds for its Betti numbers have been given in [22].Kr¨amer [21] has determined number-theoretic formulas for the numbers λ ℓ and λ ∗ ℓ of conjugacyclasses of finite subgroups in the full Bianchi groups. Kr¨amer’s formulas have been evaluatedfor hundreds of thousands of Bianchi groups [29], and these values are matching with the onesfrom the orbifold structure computations with [28] in the cases where the latter are available.When we pass to the complexified orbifold [ H C / Γ], the real line that is the rotation axisin H R of an element of finite order, becomes a complex line. However, the centralizer still actsin the same way by reflections and translations. So, the interval b b as a quotient of the realline yields a stripe b b × R as a quotient of the complex line. And the circle b as a quotientof the real line yields a cylinder b × R as a quotient of the complex line. Therefore, using thedegree shifting numbers computed in [31], we obtain the result of Theorem 9,H dorb (cid:0) [ H C / Γ] (cid:1) ∼ = H d ( H C / Γ ; Q ) ⊕ Q λ +2 λ − λ ∗ , d = 2 , Q λ − λ ∗ +2 λ − λ ∗ , d = 3 , , otherwise . eferences [1] Marian F. Anton, On a conjecture of Quillen at the prime
3, J. Pure Appl. Algebra (1999), no. 1, 1–20,DOI 10.1016/S0022-4049(98)00050-4. MR1723188 (2000m:19003)[2] ,
Homological symbols and the Quillen conjecture , J. Pure Appl. Algebra (2009), no. 4, 440–453,DOI 10.1016/j.jpaa.2008.07.011. MR2483829 (2010f:20042)[3] A. Ash, P. E. Gunnells, M. McConnell, and D. Yasaki,
On the growth of torsion in the cohomology of arith-metic groups , J. Inst. Math. Jussieu (2020), no. 2, 537–569, DOI 10.1017/s1474748018000117. MR4079152[4] Nicolas Bergeron, Mehmet Haluk S¸eng¨un, and Akshay Venkatesh, Torsion homology growth and cycle com-plexity of arithmetic manifolds , Duke Math. J. (2016), no. 9, 1629–1693, DOI 10.1215/00127094-3450429.MR3513571[5] Ethan Berkove and Alexander D. Rahm,
The mod 2 cohomology rings of SL of the imaginary quadratic in-tegers , J. Pure Appl. Algebra (2016), no. 3, 944–975, DOI 10.1016/j.jpaa.2015.08.002. With an appendixby Aurel Page. MR3414403[6] Ethan Berkove, Grant Lakeland, and Alexander D. Rahm, The mod cohomology rings of congruencesubgroups in the Bianchi groups , J. Algebr. Comb. (2019), https://doi.org/10.1007/s10801-019-00912-8 .[7] Kenneth S. Brown, Cohomology of groups , Graduate Texts in Mathematics, vol. 87, Springer-Verlag, NewYork, 1994. Corrected reprint of the 1982 original. MR1324339 (96a:20072)[8] Alexander D. Rahm and Anh Tuan Bui,
Verification of the Quillen conjecture in the rank 2 imaginaryquadratic case , Vol. 22, 2020.[9] Anh Tuan Bui and Alexander D. Rahm,
Torsion Subcomplexes package in HAP , a GAP subpackage, http://hamilton.nuigalway.ie/Hap/doc/chap26.html .[10] Anh Tuan Bui, Alexander D. Rahm, and Matthias Wendt,
On Farrell–Tate cohomology of GL(3) over ringsof quadratic integers (2020). Preprint, https://hal.archives-ouvertes.fr/hal-02435963 .[11] Weimin Chen and Yongbin Ruan,
A new cohomology theory of orbifold , Comm. Math. Phys. (2004),no. 1, 1–31. MR2104605 (2005j:57036), Zbl 1063.53091[12] Michael W. Davis,
The geometry and topology of Coxeter groups , London Mathematical Society MonographsSeries, vol. 32, Princeton University Press, Princeton, NJ, 2008. MR2360474 (2008k:20091)[13] Mathieu Dutour Sikiri´c and Philippe Elbaz-Vincent and Alexander Kupers and Jacques Martinet,
Voro-noi complexes in higher dimensions, cohomology of GL N ( Z ) for N ≥ and the triviality of K ( Z ),arXiv:1910.11598[math.KT], 2019.[14] William G. Dwyer, Exotic cohomology for GL n ( Z [1 / (1998), no. 7, 2159–2167, DOI 10.1090/S0002-9939-98-04279-8. MR1443381 (2000a:57092)[15] Graham Ellis, An invitation to computational homotopy , Oxford University Press, Oxford, 2019. MR3971587[16] Barbara Fantechi and Lothar G¨ottsche,
Orbifold cohomology for global quotients , Duke Math. J. (2003),no. 2, 197–227. MR1971293 (2004h:14062), Zbl 1086.14046[17] Hans-Werner Henn,
The cohomology of
SL(3 , Z [1 / K -Theory (1999), no. 4, 299–359. MR1683179(2000g:20087)[18] Hans-Werner Henn, Jean Lannes, and Lionel Schwartz, Localizations of unstable A -modules and equivariantmod p cohomology , Math. Ann. (1995), no. 1, 23–68, DOI 10.1007/BF01446619. MR1312569 (95k:55036)[19] Felix Klein, Ueber bin¨are Formen mit linearen Transformationen in sich selbst , Math. Ann. (1875), no. 2,183–208. MR1509857[20] Kevin P. Knudson, Homology of linear groups , Progress in Mathematics, vol. 193, Birkh¨auser Verlag, Basel,2001. MR1807154 (2001j:20070)[21] Norbert Kr¨amer,
Die Konjugationsklassenanzahlen der endlichen Untergruppen in der Norm-Eins-Gruppevon Maximalordnungen in Quaternionenalgebren , Diplomarbeit, Mathematisches Institut, Universit¨at Bonn,1980. http://tel.archives-ouvertes.fr/tel-00628809/ (German).[22] ,
Beitr¨age zur Arithmetik imagin¨arquadratischer Zahlk¨orper , Math.-Naturwiss. Fakult¨at der Rheinis-chen Friedrich-Wilhelms-Universit¨at Bonn; Bonn. Math. Schr., 1984.[23] Jean-Fran¸cois Lafont, Ivonne J. Ortiz, Alexander D. Rahm, and Rub´en J. S´anchez-Garc´ıa,
Equivari-ant K -homology for hyperbolic reflection groups , Q. J. Math. (2018), no. 4, 1475–1505, DOI10.1093/qmath/hay030. MR3908707
24] Stephen A. Mitchell,
On the plus construction for B GL Z [ ] at the prime
2, Math. Z. (1992), no. 2,205–222, DOI 10.1007/BF02570830. MR1147814 (93b:55021)[25] Fabio Perroni and Alexander D. Rahm,
On Ruan’s cohomological crepant resolution conjecture for the com-plexified Bianchi orbifolds , Algebr. Geom. Topol. (2019), no. 6, 2715–2762, DOI 10.2140/agt.2019.19.2715.MR4023327[26] Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II , Ann. of Math. (2) (1971), 549–572;ibid. (2) 94 (1971), 573–602.[27] Alexander D. Rahm, Homology and K -theory of the Bianchi groups (Homologie et K -th´eorie des groupesde Bianchi) , Comptes Rendus Math´ematique de l’ Acad´emie des Sciences - Paris (2011), no. 11-12,615–619.[28] Alexander D. Rahm, Bianchi.gp , Open source program (GNU general public license), validated by theCNRS: subject to the Certificat de Comp´etences en Cal-cul Intensif (C3I) and part of the GP scripts library of Pari/GP Development Center, 2010.[29] Alexander D. Rahm,
Accessing the cohomology of discrete groups above their virtual cohomological dimension ,J. Algebra (2014), 152–175. MR3177890[30] Alexander D. Rahm,
The homological torsion of
PSL of the imaginary quadratic integers , Trans. Amer.Math. Soc. (2013), no. 3, 1603–1635. MR3003276[31] Alexander D. Rahm, On the equivariant K -homology of PSL of the imaginary quadratic integers , Annalesde l’Institut Fourier (2016), no. 4, 1667–1689, http://dx.doi.org/10.5802/aif.3047 .[32] Alexander D. Rahm, Higher torsion in the Abelianization of the full Bianchi groups , LMS J. Comput. Math. (2013), 344–365. MR3109616[33] Anh Tuan Bui, Alexander D. Rahm, and Matthias Wendt, The Farrell-Tate and Bredon homo-logy for
PSL ( Z ) via cell subdivisions , J. Pure Appl. Algebra (2019), no. 7, 2872–2888, DOI10.1016/j.jpaa.2018.10.002. MR3912952[34] Alexander D. Rahm and Mathias Fuchs, The integral homology of
PSL of imaginary quadratic integers withnon-trivial class group , J. Pure Appl. Algebra (2011), no. 6, 1443–1472, DOI 10.1016/j.jpaa.2010.09.005.Zbl 1268.11072[35] Alexander D. Rahm and Matthias Wendt, On Farrell–Tate cohomology of SL over S -integers , Vol. 512,2018. MR3841530[36] Alexander D. Rahm and Matthias Wendt, A refinement of a conjecture of Quillen , ComptesRendus Math´ematique de l’Acad´emie des Sciences (2015), no. 9, 779–784, DOIhttp://dx.doi.org/10.1016/j.crma.2015.03.022.[37] Alexander Scheutzow,
Computing rational cohomology and Hecke eigenvalues for Bianchi groups , J. NumberTheory (1992), no. 3, 317–328, DOI 10.1016/0022-314X(92)90004-9. MR1154042 (93b:11068)[38] Joachim Schwermer and Karen Vogtmann, The integral homology of SL and PSL of Euclidean imaginaryquadratic integers , Comment. Math. Helv. (1983), no. 4, 573–598, DOI 10.1007/BF02564653. MR728453(86d:11046)[39] Jean-Pierre Serre, Cohomologie des groupes discrets , Prospects in mathematics (Proc. Sympos., PrincetonUniv., Princeton, N.J., 1970), Princeton Univ. Press, Princeton, N.J., 1971, pp. 77–169. Ann. of Math.Studies, No. 70 (French). MR0385006[40] Christophe Soul´e,
The cohomology of SL ( Z ), Topology (1978), no. 1, 1–22.[41] Karen Vogtmann, Rational homology of Bianchi groups , Math. Ann. (1985), no. 3, 399–419. MR799670(87a:22025), Zbl 0545.20031[42] C. Terence C. Wall,
Resolutions for extensions of groups , Proc. Cambridge Philos. Soc. (1961), 251–255.MR0178046 (31 Homology of SL over function fields I: parabolic subcomplexes , J. Reine Angew. Math. (2018), 159–205, DOI 10.1515/crelle-2015-0047. MR3808260 Universit´e de la Polyn´esie Franc¸aise, Laboratoire GAATI