Equivariant K-homology for hyperbolic reflection groups
Jean-François Lafont, Ivonne J. Ortiz, Alexander Rahm, Rubén J. Sánchez-García
EEQUIVARIANT K-HOMOLOGY FOR HYPERBOLICREFLECTION GROUPS
JEAN-FRANC¸ OIS LAFONT, IVONNE J. ORTIZ, ALEXANDER D. RAHM,AND RUB´EN J. S ´ANCHEZ-GARC´IA
Abstract.
We compute the equivariant K -homology of the classifying spacefor proper actions, for cocompact 3-dimensional hyperbolic reflection groups.This coincides with the topological K -theory of the reduced C ∗ -algebra as-sociated to the group, via the Baum-Connes conjecture. We show that, forany such reflection group, the associated K -theory groups are torsion-free. Asa result we can promote previous rational computations to integral compu-tations. Our proof relies on a new efficient algebraic criterion for checkingtorsion-freeness of K -theory groups, which could be applied to many otherclasses of groups. Introduction
For a discrete group Γ, a general problem is to compute K ∗ ( C ∗ r Γ), the topological K -theory of the reduced C ∗ -algebra of Γ. The Baum-Connes Conjecture predictsthat this functor can be determined, in a homological manner, from the complexrepresentation rings of the finite subgroups of Γ. This viewpoint led to generalrecipes for computing the rational topological K -theory K ∗ ( C ∗ r Γ) ⊗ Q of groups,through the use of Chern characters (see for instance L¨uck and Oliver [17] andL¨uck [14], [16], as well as related earlier work of Adem [1]). When Γ has smallhomological dimension, one can sometimes even give completely explicit formulasfor the rational topological K -theory, see for instance Lafont, Ortiz, and S´anchez-Garc´ıa [12] for the case where Γ is a 3-orbifold group.On the other hand, performing integral calculations for these K -theory groupsis much harder. For 2-dimensional crystallographic groups, such calculations havebeen done in M. Yang’s thesis [26]. This was subsequently extended to the class ofcocompact planar groups by L¨uck and Stamm [19], and to certain higher dimen-sional dimensional crystallographic groups by Davis and L¨uck [5] (see also Langerand L¨uck [13]). For 3-dimensional groups, L¨uck [15] completed this calculationfor the semi-direct product Hei ( Z ) (cid:111) Z of the 3-dimensional integral Heisenberggroup with a specific action of the cyclic group Z . Some further computations werecompleted by Isely [8] for groups of the form Z (cid:111) Z ; by Rahm [23] for the classof Bianchi groups; by Pooya and Valette [22] for solvable Baumslag-Solitar groups;and by Flores, Pooya and Valette [6] for lamplighter groups of finite groups.Our present paper has two main goals. Our first goal is to add to the list ofexamples, by providing a formula for the integral K -theory groups of cocompact3-dimensional hyperbolic reflection groups. Date : April 4, 2019. a r X i v : . [ m a t h . K T ] A p r LAFONT, ORTIZ, RAHM, AND S´ANCHEZ-GARC´IA
Main Theorem.
Let Γ be a cocompact -dimensional hyperbolic reflection group,generated by reflections in the side of a hyperbolic polyhedron P ⊂ H . Then K ( C ∗ r (Γ)) ∼ = Z cf (Γ) and K ( C ∗ r (Γ)) ∼ = Z cf (Γ) − χ ( C ) , where the integers cf (Γ) , χ ( C ) can be explicitly computed from the combinatorics ofthe polyhedron P . Here, cf (Γ) denotes the number of conjugacy classes of elements of finite orderin Γ, and χ ( C ) denotes the Euler characteristic of the Bredon chain complex. By acelebrated result of Andre’ev [2], there is a simple algorithm that inputs a Coxetergroup Γ, and decides whether or not there exists a hyperbolic polyhedron P Γ ⊂ H which generates Γ. In particular, given an arbitrary Coxeter group, one can easilyverify if it satisfies the hypotheses of our Main Theorem.Note that the lack of torsion in the K -theory is not a property shared by alldiscrete groups acting on hyperbolic 3-space. For example, 2-torsion occurs in K ( C ∗ r (Γ)) whenever Γ is a Bianchi group containing a 2-dihedral subgroup C × C (see [23]). In fact, the key difficulty in the proof of our Main Theorem lies in showingthat these K -theory groups are torsion-free. Some previous integral computationsyielded K -theory groups that are torsion-free, though in those papers the torsion-freeness was a consequence of ad-hoc computations. Our second goal is to give ageneral criterion which explains the lack of torsion, and can be efficiently checked.This allows a systematic, algorithmic approach to the question of whether a K -theory group is torsion-free.Let us briefly describe the contents of the paper. In Section 2, we providebackground material on hyperbolic reflection groups, topological K -theory, and theBaum-Connes Conjecture. We also introduce our main tool, the Atiyah-Hirzebruchtype spectral sequence. In Section 3, we use the spectral sequence to show that the K -theory groups we are interested in coincide with the Bredon homology groupsH Fin (Γ; R C ) and H Fin (Γ; R C ) respectively. We also explain, using the Γ-action on H , why the homology group H Fin (Γ; R C ) is torsion-free. In contrast, showing thatH Fin (Γ; R C ) is torsion-free is much more difficult. In Section 4, we give a geometricproof for this fact in a restricted setting. In Section 5, we give a linear algebraicproof in the general case, inspired by the “representation ring splitting” techniqueof [23]. In particular, we establish a novel criterion (Theorem 5) for verifying thatH Fin (Γ; R C ) is torsion-free, for any collection of groups Γ with prescribed typesof finite subgroups. In Sections 6.1 and 6.2, we further illustrate this criterion byapplying it to the Heisenberg semidirect product group of [15] and to the crystallo-graphic groups of [5] respectively. As the rank of the K -theory groups can be easilycomputed, this gives an alternate proof of the integral K -theoretic computationsin [15] and [5]. Finally, in Section 7, we return to our Coxeter groups, and providean explicit formula for the rank of the Bredon homology groups (and hence for the K -groups we are interested in), in terms of the combinatorics of the polyhedron P .Our paper concludes with two fairly long appendices, which contain all the char-acter tables and induction homomorphisms used in our proofs, making our resultsexplicit and self-contained. QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 3
Acknowledgments
Portions of this work were carried out during multiple collaborative visits atOhio State University, Miami University, and the University of Southampton. Theauthors would like to thank these institutions for their hospitality. The authorswould also like to thank the anonymous referee for several helpful suggestions.Lafont was partly supported by the NSF, under grant DMS-1510640. Ortiz waspartly supported by the NSF, under grant DMS-1207712. Rahm was supported byGabor Wiese’s University of Luxembourg grant AMFOR.2.
Background Material K -theory and the Baum-Connes Conjecture. Associated to a discretegroup Γ, one has C ∗ r Γ, the reduced C ∗ -algebra of Γ. This algebra is defined to bethe closure, with respect to the operator norm, of the linear span of the imageof the regular representation λ : Γ → B ( l (Γ)) of Γ on the Hilbert space l (Γ)of square-summable complex valued functions on Γ. This algebra encodes variousanalytic properties of the group Γ [21].For a C ∗ -algebra A , one can define the topological K -theory groups K ∗ ( A ) := π ∗− ( GL ( A )), which are the homotopy groups of the space GL ( A ) of invertible ma-trices with entries in A . Due to Bott periodicity, there are canonical isomorphisms K ∗ ( A ) ∼ = K ∗ +2 ( A ), and thus it is sufficient to consider K ( A ) and K ( A ).In the special case where A = C ∗ r Γ, the
Baum-Connes Conjecture predicts thatthere is a canonical isomorphism K Γ n ( X ) → K n ( C ∗ r (Γ)), where X is a model for E Γ(the classifying space for Γ-actions with isotropy in the family of finite subgroups),and K Γ ∗ ( − ) is the equivariant K -homology functor. The Baum-Connes conjecturehas been verified for many classes of groups. We refer the interested reader to themonograph by Mislin and Valette [21], or the survey article by L¨uck and Reich [18]for more information on these topics.2.2. Hyperbolic reflection groups.
We will assume some familiarity with thegeometry and topology of Coxeter groups, which the reader can obtain from Davis’book [4]. By a d -dimensional hyperbolic polyhedron , we mean a bounded region ofhyperbolic d -space H d delimited by a given finite number of (geodesic) hyperplanes,that is, the intersection of a collection of half-spaces associated to the hyperplanes.Let P ⊂ H d be a polyhedron such that all the interior angles between intersectingfaces are of the form π/m ij , where the m ij ≥ P be the associated Coxeter group, generated byreflections in the hyperplanes containing the faces of P .The Γ-space H d is then a model for E Γ, with fundamental domain P . Thisis a strict fundamental domain – no further points of P are identified under thegroup action – and hence P = Γ \ H d . Recall that Γ admits the following Coxeterpresentation:(1) Γ = (cid:104) s , . . . , s n | ( s i s j ) m ij (cid:105) , where n is the number of distinct hyperplanes enclosing P , s i denotes the reflectionon the i th face, and m ij ≥ m ii = 1 for all i , and, if i (cid:54) = j ,the corresponding faces meet with interior angle π/m ij . We will write m ij = ∞ ifthe corresponding faces do not intersect. For the rest of this article, d = 3, and X is H with the Γ-action described above, with fundamental domain P . LAFONT, ORTIZ, RAHM, AND S´ANCHEZ-GARC´IA
Cell structure of the orbit space.
Fix an ordering of the faces of P withindexing set J = { , . . . , n } . We will write (cid:104) S (cid:105) for the subgroup generated by asubset S ⊂ Γ. At a vertex of P , the concurrent faces (a minimum of three) mustgenerate a reflection group acting on the 2-sphere, hence it must be a spherical tri-angle group. This forces the number of incident faces to be exactly three. The onlyfinite Coxeter groups acting by reflections on S are the triangle groups ∆(2 , , m )for some m ≥
2, ∆(2 , , , ,
4) and ∆(2 , , p, q, r ) = (cid:10) s , s , s | s , s , s , ( s s ) p , ( s s ) q , ( s s ) r (cid:11) . From our compact polyhedron P , we obtain an induced Γ-CW-structure on X = H with: • one orbit of 3-cells, with trivial stabilizer; • n orbits of 2-cells (faces) with stabilizers (cid:104) s i | s i (cid:105) ∼ = Z / i = 1 , . . . , n ); • one orbit of 1-cells (edges) for each unordered pair i, j ∈ J with m ij (cid:54) = ∞ ,with stabilizer a dihedral group D m ij — this group structure can be readoff straight from the Coxeter presentation (cid:104) s i , s j | s i , s j , ( s i s j ) m ij (cid:105) ; • one orbit of 0-cells (vertices) per unordered triple i, j, k ∈ J with (cid:104) s i , s j , s k (cid:105) finite, with stabilizer the triangle group (cid:104) s i , s j , s k (cid:105) ∼ = ∆( m ij , m ik , m jk ).We introduce the following notation for the simplices of P : f i (faces), e ij = f i ∩ f j (edges) , (3) v ijk = f i ∩ f j ∩ f k = e ij ∩ e ik ∩ e jk (vertices) , whenever the intersections are non-empty.2.4. A spectral sequence.
We ultimately want to compute the K -theory groupsof the reduced C ∗ -algebra of Γ via the Baum-Connes conjecture. Note that theconjecture holds for these groups: Coxeter groups have the Haagerup property [3]and hence satisfy Baum-Connes [7]. Therefore, it suffices to compute the equivari-ant K -homology groups K Γ ∗ ( X ), since X is a model of E Γ. In turn, these groupscan be obtained from the Bredon homology of X , calculated via an equivariantAtiyah-Hirzebruch spectral sequence coming from the skeletal filtration of the Γ-CW-complex X (cf. [20]). The second page of this spectral sequence is given bythe Bredon homology groups(4) E p,q = (cid:40) H Fin p (Γ; R C ) q even , q odd . The coefficients R C of the Bredon homology groups are given by the complex repre-sentation ring of the cell stabilizers, which are finite subgroups. In order to simplifynotation, we will often write H p to denote H Fin p (Γ; R C ). In our case dim( X ) = 3, sothe Atiyah-Hirzebruch spectral sequence is particularly easy to analyze, and givesthe following: Proposition 1.
There are short exact sequences (cid:47) (cid:47) coker( d , ) (cid:47) (cid:47) K Γ0 ( X ) (cid:47) (cid:47) H (cid:47) (cid:47) and (cid:47) (cid:47) H (cid:47) (cid:47) K Γ1 ( X ) (cid:47) (cid:47) ker( d , ) (cid:47) (cid:47) , QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 5 where d , : E , = H −→ E , = H is the differential on the E -page of theAtiyah-Hirzebruch spectral sequence.Proof. This follows at once from a result of Mislin [20, Theorem 5.29]. (cid:3)
Bredon Homology.
To lighten the notation, we write Γ e for the stabilizerin Γ of the cell e . The Bredon homology groups in Equation (4) are defined to bethe homology groups of the following chain complex:(5) . . . (cid:47) (cid:47) (cid:76) e ∈ I d R C (Γ e ) ∂ d (cid:47) (cid:47) (cid:76) e ∈ I d − R C (Γ e ) (cid:47) (cid:47) . . . , where I d is a set of orbit representatives of d -cells ( d ≥
0) in X . The differentials ∂ d are defined via the geometric boundary map and induction between representationrings. More precisely, if ge (cid:48) is in the boundary of e ( e ∈ I d , e (cid:48) ∈ I d − , g ∈ Γ), then ∂ restricted to R C (Γ e ) → R C (Γ e (cid:48) ) is given by the composition R C (Γ e ) ind (cid:47) (cid:47) R C (Γ ge (cid:48) ) ∼ = (cid:47) (cid:47) R C (Γ e (cid:48) ) , where the first map is the induction homomorphism of representation rings asso-ciated to the subgroup inclusion Γ e ⊂ Γ ge (cid:48) , and the second is the isomorphisminduced by conjugation Γ ge (cid:48) = g Γ e (cid:48) g − . Finally, we add a sign depending on achosen (and thereafter fixed) orientation on the faces of P . The value ∂ d ( e ) equalsthe sum of these maps over all boundary cells of e .Since P is a strict fundamental domain, we can choose the faces of P as orbitrepresentatives. With this choice of orbit representatives, the g in the previousparagraph is always the identity. We will implicitly make this assumption fromnow on. 3. Analyzing the Bredon chain complex for
ΓLet S = { s i : 1 ≤ i ≤ n } be the set of Coxeter generators and J = { , . . . , n } .Since X is 3-dimensional, the Bredon chain complex associated to X reduces to(6) 0 (cid:47) (cid:47) C ∂ (cid:47) (cid:47) C ∂ (cid:47) (cid:47) C ∂ (cid:47) (cid:47) C (cid:47) (cid:47) . We now have to analyze the differentials in the above chain complex. Recall that fora finite group G , the complex representation ring R C ( G ) is defined as the free abeliangroup with basis the set of irreducible representations of G (the ring structure isnot relevant in this setting). Hence R C ( G ) ∼ = Z c ( G ) , where we write c ( G ) for theset of conjugacy classes in G .3.1. Analysis of ∂ . Let G be a finite group with irreducible representations ρ , . . . , ρ m of degree n , . . . , n m , and τ the only representation of the trivial sub-group { G } ≤ G . Then τ induces the regular representation of G :(7) Ind G { G } ( τ ) = n ρ + . . . + n m ρ m . Lemma 1.
Let X be a Γ -CW-complex with finite stabilizers, and k ∈ N . If there isa unique orbit of k -cells and this orbit has trivial stabilizer, then H k = 0 , providedthat ∂ k (cid:54) = 0 .Proof. The Bredon module C k equals R C ( (cid:104) (cid:105) ) ∼ = Z with generator τ , the trivial rep-resentation. Then ∂ k ( τ ) (cid:54) = 0 implies ker( ∂ k ) = 0; and therefore the correspondinghomology group vanishes. (cid:3) LAFONT, ORTIZ, RAHM, AND S´ANCHEZ-GARC´IA
From the lemma, we can easily see that H = 0 if ∂ (cid:54) = 0. Indeed, for ∂ = 0 tooccur, one would need all boundary faces of P to be pairwise identified. This cannothappen since P is a strict fundamental domain – the group acts by reflections onthe faces. The Lemma then forces H = 0, and Proposition 1 gives us: Corollary 1.
We have K Γ1 ( X ) ∼ = H , and there is a short exact sequence (cid:47) (cid:47) H (cid:47) (cid:47) K Γ0 ( X ) (cid:47) (cid:47) H (cid:47) (cid:47) . Analysis of ∂ . Let f be a face of P and e ∈ ∂f an edge. Suppose, usingthe notation of Equation (3), that f = f i and e = e ij . Then we have a map R C ( (cid:104) s i (cid:105) ) → R C ( (cid:104) s i , s j (cid:105) ) induced by inclusion. Recall that (cid:104) s i (cid:105) ∼ = C and (cid:104) s i , s j (cid:105) ∼ = D m ij . Denote the characters of these two finite groups as specified in Tables 1and 2; and denote by a character name with the suffix “ ↑ ” the character inducedin the ambient larger group. C e s i ρ ρ − Table 1.
Character table of (cid:104) s i (cid:105) ∼ = C . D m ( s i s j ) r s j ( s i s j ) r χ χ − (cid:99) χ ( − r ( − r (cid:99) χ ( − r ( − r +1 φ p (cid:0) πprm (cid:1) Table 2.
Character table of (cid:104) i , s j (cid:105) ∼ = D m , where i < j and 0 ≤ r ≤ m −
1, while 1 ≤ p ≤ m/ − m is even, and 1 ≤ p ≤ ( m − / m is odd, and where the hat (cid:98) denotes a character which appearsonly when m is even.A straightforward analysis (provided in Section B.2 of Appendix B) shows that ρ ↑ = χ + (cid:99) χ + (cid:88) φ p ,ρ ↑ = χ + (cid:99) χ + (cid:88) φ p , if i < j , or ρ ↑ = χ + (cid:99) χ + (cid:88) φ p ,ρ ↑ = χ + (cid:99) χ + (cid:88) φ p , if j < i . Thus the induction map on the representation rings is the morphism offree abelian groups Z → Z c ( D mij ) given by( a, b ) (cid:55)→ ± ( a, b, (cid:98) b, (cid:98) a, a + b, . . . , a + b ) or( a, b ) (cid:55)→ ± ( a, b, (cid:98) a, (cid:98) b, a + b, . . . , a + b ) , QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 7 where again the hat (cid:98) denotes an entry which appears only when m ij is even. Usingthe analysis above, we can now show the following. Theorem 1. If P is compact, then H = 0 . From this theorem and Corollary 1, we immediately obtain:
Corollary 2. K Γ0 ( X ) = H and K Γ1 ( X ) = H .Proof of Theorem 1. Fix an orientation on the polyhedron P , and consider theinduced orientations on the faces. At an edge we have two incident faces f i and f j with opposite orientations. So without loss of generality we have, as a map of freeabelian groups, R C ( (cid:104) s i (cid:105) ) ⊕ R C ( (cid:104) s j (cid:105) ) → R C ( (cid:104) s i , s j (cid:105) ) , (8) ( a i , b i | a j , b j ) (cid:55)→ ( a i − a j , b i − b j , (cid:92) a i − b j , (cid:92) b i − a j , S, . . . , S ) , where S = a i + b i − a j − b j , and the elements with a hat (cid:98) appear only when m ij is even. Note that we use vertical bars ‘ | ’ for clarity, to separate elements comingfrom different representation rings.By the preceding analysis, ∂ ( x ) = 0 implies that, for each i, j ∈ J such that thecorresponding faces f i and f j meet, we have(1) a i = a j and b i = b j , and(2) a i = a j = b i = b j , if m ij is even.Suppose that f , . . . , f n are the faces of P . Let x = ( a , b | . . . | a n , b n ) ∈ C bean element in Ker( ∂ ). Note that ∂ P is connected (since P is homeomorphic to D ), so by (1) and (2) above, we have that a = . . . = a n and b = . . . = b n .Since the stabilizer of a vertex is a spherical triangle group, there is an even m ij ,which also forces a = b . Therefore, we have x = ( a, a | . . . | a, a ), so x = ∂ ( a ) (notethat the choice of orientation above forces all signs to be positive). This yieldsker( ∂ ) ⊆ im( ∂ ), which gives the vanishing of the second homology group. (cid:3) Analysis of ∂ . Let e = e ij be an edge and v = v ijk ∈ ∂e a vertex, usingthe notation of Equation (3). We study all possible induction homomorphisms R C ( (cid:104) s i , s j (cid:105) ) → R C ( (cid:104) s i , s j , s k (cid:105) ) in Appendix B, and in this section use these com-putations to verify that H is torsion-free. Theorem 2.
There is no torsion in H .Proof. Consider the Bredon chain complex C ∂ (cid:47) (cid:47) C ∂ (cid:47) (cid:47) C . To prove that H = ker( ∂ ) / im( ∂ ) is torsion-free, it suffices to prove that C / im( ∂ )is torsion-free. Let α ∈ C and 0 (cid:54) = k ∈ Z such that kα ∈ im( ∂ ). We shall provethat α ∈ im( ∂ ).Since kα ∈ im( ∂ ), we can find β ∈ C with ∂ ( β ) = kα . Suppose that P has n faces, and write β = ( a , b | . . . | a n , b n ) ∈ C , using vertical bars ‘ | ’ to separateelements coming from different representation rings. We shall see that one can finda 1-chain β (cid:48) , homologous to β , and with every entry of β (cid:48) a multiple of k .At an edge e ij , the differential ∂ is described in Equation (8). Since every entryof ∂ ( β ) is a multiple of k , we conclude from Equation (8) that for every pair ofintersecting faces f i and f j , a i ≡ a j (mod k ) and b i ≡ b j (mod k ) . LAFONT, ORTIZ, RAHM, AND S´ANCHEZ-GARC´IA
Equation (8) also shows that ∂P = (1 , | . . . | , ∂P ∈ C , is in the kernelof ∂ . In particular, setting β (cid:48) = β − a ∂P , we see that ∂ ( β (cid:48) ) = ∂ ( β ) = α andwe can assume without loss of generality that β (cid:48) satisfies a (cid:48) ≡ k ).Let us consider the coefficients for the 1-chain β (cid:48) . For every face f j intersecting f , we have a (cid:48) − a (cid:48) j ≡ k ), which implies a (cid:48) j ≡ k ). Since ∂ P isconnected, repeating this argument we arrive at a (cid:48) i ≡ k ) for all i . Inaddition, there are even m ij (the stabilizer of a vertex is a spherical triangle group),and hence (8) also yields a (cid:48) i − b (cid:48) j ≡ k ), which implies b (cid:48) j ≡ k ).Exactly the same argument as above then ensures that b (cid:48) i ≡ k ) for all i .Since all coefficients of β (cid:48) are divisible by k , we conclude that α = ∂ ( β (cid:48) /k ) ∈ im( ∂ ), as desired. (cid:3) We note that a similar method of proof can be used, in many cases, to show that H is torsion-free – though the argument becomes much more complicated. Thisapproach is carried out in Section 4. Corollary 3.
Let cf (Γ) be the number of conjugacy classes of elements of finiteorder in Γ , and χ ( C ) the Euler characteristic of the Bredon chain complex (6).Then we have H ∼ = Z cf (Γ) − χ ( C ) . Proof.
The Euler characteristic of a chain complex coincides with the alternatingsum of the ranks of the homology groups, giving us χ ( C ) = rank( H ) − rank( H ) + rank( H ) − rank( H ) . Since H = H = 0, we have rank( H ) = rank( H ) − χ ( C ), and rank( H ) = cf (Γ) [20]. Since H is torsion-free (Theorem 2), the result follows. (cid:3) Note that both cf (Γ) and χ ( C ) can be obtained directly from the geometry ofthe polyhedron P or, equivalently, from the Coxeter integers m ij . We discuss thisfurther, and give explicit formulas, in Section 7. Remark . A previous article by three of the authors [12] gave an algorithm tocompute the rank of the Bredon homology for groups Γ with a cocompact, 3-manifold model X for the classifying space E Γ. The interested reader can easilycheck that the computations in the present paper agree with the calculations in [12].To complete the computation of the Bredon homology, and hence of the equi-variant K -homology, all that remains is to compute the torsion subgroup of H .We will show that in fact H is also torsion-free. Theorem 3.
There is no torsion in H . We postpone the proof to Section 5 below. We note the following immediateconsequence of Theorem 3.
Corollary 4. K Γ0 ( X ) is torsion-free of rank cf (Γ) . Our Main Theorem now follows immediately by combining Corollaries 2, 3, and 4.Moreover, in Section 7, we will give a formula for cf (Γ) and χ ( C ) from the combi-natorics of the polyhedron. QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 9 No torsion in H – the geometric approach We present a geometric proof for a restricted version of Theorem 3. The methodof proof is similar to the proof of Theorem 2, but with further technical difficulties.We will show:
Theorem 4.
Assume the compact polyhedron P is such that all vertex stabilizersare of the form D m × Z , where m ≥ can vary from vertex to vertex. Then thereis no torsion in the -dimensional Bredon homology group H Fin (Γ; R C ) . First, we discuss some terminology and the overall strategy of the proof. Fix k ≥ k -torsion in H . Let β ∈ C (inthe Bredon chain complex of Γ) such that ∂ ( β ) is the zero vector modulo k . Notethat an element α ∈ C has order k in H = C / im( ∂ ) if and only if kα = ∂ ( β ).Recall that C = (cid:77) e ij edge R C ( (cid:104) s i , s j (cid:105) ) , is the direct sum of the representation rings (as abelian groups) of the edge stabiliz-ers. The coefficients of x supported along a particular edge e ij is by definition theprojection of x to Z n ij ∼ = R C ( (cid:104) s i , s j (cid:105) ), where n ij is the dimension of the represen-tation ring of the edge stabilizer. We say an element x is k -divisible along an edge e ij provided the coefficients of x supported along e ij are congruent to zero mod k .By establishing k -divisibility of β along an edge e ij , we mean: substituting β by a homologous element β (cid:48) ∈ C (homologous means that ∂ ( β ) = ∂ ( β (cid:48) )), suchthat β (cid:48) is k -divisible along e ij . When the 1-chain β is clear from the context, wewill abuse terminology and simply say that the edge e is k -divisible. We sometimesrefer to an edge with stabilizer D n as an n -edge , or edge of type n .Our goal is to replace β with a homologous chain β (cid:48) for which all the edges are k -divisible, that is,(1) ∂ ( β (cid:48) ) = ∂ ( β ), and(2) every coefficient in the 1-chain β (cid:48) is divisible by k .If we can do this, it follows that α = ∂ ( β (cid:48) /k ), and hence that α is zero in H .The construction of β (cid:48) is elementary, but somewhat involved. It proceeds via aseries of steps, which will be described in the following subsections. Sections 4.1 to4.9 contain the conceptual, geometric arguments needed for the proof. At severalsteps in the proof, we require some technical algebraic lemmas. For the sake ofexposition, we defer these lemmas and their proofs to the very last Section 4.11.4.1. Coloring the -skeleton. Recall that β ∈ C = (cid:76) e ∈P (1) R C (Γ e ), so we canview β as a formal sum of complex representations of the stabilizers of the variousedges in the 1-skeleton of P . The edge stabilizers are dihedral groups. Let us 2-colorthe edges of the polyhedron, blue if the stabilizer is D , and red if the stabilizer is D m , where m ≥
3. From our constraints on the vertex groups, we see that everyvertex has exactly two incident blue edges. Of course, any graph with all verticesof degree 2 decomposes as a disjoint union of cycles.The collection of blue edges thus forms a graph, consisting of pairwise disjointloops, separating the boundary of the polyhedron P (topologically a 2-sphere) into afinite collection of regions, at least two of which must be contractible. The red edgesappear in the interior of these individual regions, joining pairs of vertices on theboundary of the region. Fixing one such contractible region R ∞ , the complement will be planar. We will henceforth fix a planar embedding of this complement. Thisallows us to view all the remaining regions as lying in the plane R . R R R R R Figure 1.
Example of enumeration of regions (see Section 4.2).4.2.
Enumerating the regions.
Our strategy for modifying β is as follows. Wewill work region by region. At each stage, we will modify β by only changing it onedges contained in the closure of a region. In order to do this, we need to enumeratethe regions.We have already identified the (contractible) region R ∞ – this will be the lastregion dealt with. In order to decide the order in which we will deal with theremaining regions, we define a partial ordering on the set of regions. For distinctregions R, R (cid:48) , we write
R < R (cid:48) if and only if R is contained in a bounded componentof R \ R (cid:48) . This defines a partial ordering on the finite set of regions. For example,any region which is minimal with respect to this ordering must be simply connected(hence contractible). We can thus enumerate the regions R , R , . . . so that, for any i < j , we have R j (cid:54) < R i (Figure 1). We will now deal with the regions in the orderthey are enumerated. Concretely, the choice of enumeration means that by thetime we get to the i th region, we have already dealt with all the regions which are“interior” to R i (i.e. in the bounded components of R \ R i ).4.3. Enumerating faces within a region.
We now want to establish k -divisibilityof the red edges inside a fixed region R . To do this, we first need to order the 2-facesinside R . Consider the graph G dual to the decomposition of R into 2-faces. Thisgraph has one vertex for each 2-face in R , and an edge joining a pair of vertices ifthe corresponding 2-faces share a (necessarily red) edge. Notice that every vertexof R lies on a pair of blue edges. It follows that, if we remove the blue edges fromthe region R , the result deformation retracts to G .If s denotes the number of regions R i < R with ∂R ∩ ∂R i non-empty, then thefirst homology H ( G ) = H ( R ) will be free abelian of rank s . We now choose anyspanning tree T for the graph G , and note that G \ T consists of precisely s edges,each of which is dual to a red edge inside the region R . Since T is a tree, we canenumerate its vertices v , v , . . . , v n so that(1) v is a leaf of the tree (vertex of degree one), and(2) the subgraph induced by any { v , . . . , v k } , 1 ≤ k ≤ n , is connected. QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 11 F F F F F F F F F F F F F v v v v v v v v v v v v v Figure 2.
Contractible region, its dual tree T with an enumera-tion of its vertices, and the corresponding enumeration of the facesin the region (see Section 4.3).Since vertices of the dual tree correspond to 2-faces in R , this gives us an enumera-tion of the 2-faces F , F , . . . , F n inside the region R . Figure 2 illustrates this processfor a contractible region, while Figure 3 gives an illustration for a non-contractibleregion. F F F F F F F F F F F F F F F F F v v v v v v v v v v v v v v v v v Figure 3.
Non-contractible region and its dual graph G . A span-ning tree T for G consisting of all edges except ( v , v ) and( v , v ). The enumeration of the vertices, and corresponding enu-meration of the faces is done according to Section 4.3.4.4. Establishing k -divisibility of red edges dual to a spanning tree. Con-tinuing to work within a fixed region R , we now explain how to establish k -divisibility of all the red edges which are dual to the edges in the spanning tree T . As explained in Section 4.3, we have an enumeration F , F , . . . of the 2-facescontained inside the region R . With our choice of enumeration, we have guar-anteed that each F k +1 shares precisely one distinguished red edge with (cid:83) ki =1 F i , distinguished in the sense that this red edge is dual to the unique edge in the tree T connecting the vertex v k +1 to the subtree spanned by v , . . . , v k .We orient the (blue) edges along the boundary loops of R clockwise, and the rededges cutting through R in an arbitrary manner. For each 2-face F i , we want tochoose corresponding elements η i in the representation ring R C (Γ F i ) ∼ = Z , whereΓ F i ∼ = C is the stabilizer of F i . These η i shall be chosen such that all red edgesdual to T are k -divisible for the 1-chain β + ∂ (cid:0) (cid:80) η i ), which is clearly homologousto β .Pick an η arbitrarily. We now assume that η , . . . , η k are given, and explainhow to choose η k +1 . By our choice of enumeration of vertices, the vertex v k +1 isadjacent to some v j where j ≤ k . Dual to the two vertices v j , v k +1 we have a pair F j , F k +1 of 2-faces inside the region R . Dual to the edge that joins v j to v k +1 is the(red) edge F j ∩ F k +1 . We see that ∂ ( η j + η k +1 ) is the only term which can changethe portion of β supported on the edge F j ∩ F k +1 . Since η j is already given, wewant to choose η k +1 in order to ensure that the resulting 1-chain is k -divisible onthe edge F j ∩ F k +1 . This will arrange key property (2) for the (red) edge F j ∩ F k +1 .That such an η k +1 can be chosen is the content of Lemma 3 in Section 4.11.Iterating this process, we find that the 1-chain β + ∂ (cid:0) (cid:80) η i ) is homologous to β ,and that all red edges dual to edges in T are k -divisible for β + ∂ (cid:0) (cid:80) η i ). R R R R R γ γ γ γ γ v v v v v Figure 4. (Left) Non-contratible region R with blue loops andred edges. (Right) Graph B associated to R . (See Section 4.5).4.5. Forming the graph B . Performing the process in Section 4.4 for each region R (including the region R ∞ ), we finally obtain a 1-chain homologous to our original β (which by abuse of notation we will still denote β ) which is k -divisible exceptpossibly along: • s i red edges inside each region R i , where s i denotes the number of regionsentirely enclosed by the region R i who share a boundary with R i ; • all the blue edges inside the 1-skeleton, which we recall decompose into afinite collection of blue loops.We now use these to form a graph B , which captures all the remaining potentiallybad edges for the chain β , i.e. edges that are still not k -divisible. B is formed withone vertex for each blue loop. Note that each remaining red edge that is potentiallynot k -divisible joins a pair of vertices which lie on some blue loops γ , γ (as all QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 13 v v v v v Figure 5.
Enumeration of vertices in B corresponding to enumer-ation of regions in Figure 1 (see Section 4.5).vertices lie on blue loops). For each such red edge, we define an edge in the graph B joining the two vertices v i corresponding to the blue loops γ i (Figure 4). Lemma 2.
The graph B is a tree.Proof. It follows immediately from the discussion in Section 4.3 that B is connected.Thus it suffices to show B has no embedded cycles. Observe that, since blue loopsseparate the plane into two connected components, the corresponding vertex of B likewise partitions the graph B into two connected components (corresponding tothe “interior” component and the “exterior” component determined by the blueloop). Thus if there is an embedded cycle, then for each vertex v of B , it mustremain within one connected component of B \{ v } . This means that any embeddedcycle has the property that all the vertices it passes through correspond to bluecomponents lying in the closure of a single region R (and all red edges lie insidethat region R ).Now by way of contradiction, assume that e , . . . , e k forms a cycle in B , cyclicallyjoining vertices γ , . . . , γ k , where the γ i are blue loops inside the closure of theregion R . We can concatenate the corresponding (red) edges e i along with pathson the blue loops γ i to obtain an embedded edge loop η contained in the closureof the region R . Now pick a pair of 2-faces F , F inside the region R , where F iscontained inside the loop η , while F is contained outside of the loop η , e.g. pickthe 2-faces on either side of the red edge e . Note that, by construction, the closedloop η separates these regions from each other.These two regions correspond to vertices v , v in the graph G associated to theregion R . Since T was a spanning tree for the graph G , it follows that we can finda sequence of edges in the tree T connecting v to v . This gives rise to a sequenceof 2-faces connecting F to F , where each consecutive face share an edge distinctfrom any of the red edges e i . Thus we obtain a continuous path joining F to F which is completely disjoint from η , a contradiction. We conclude that B cannotcontain any cycles, and hence is a tree. (cid:3) Notice that each vertex in B corresponds to a blue loop, which lies in the closureof precisely two regions. There will thus be a unique such region which lies in thebounded component of the complement of the loop. This establishes a bijectionbetween the regions and the vertices of B . From the enumeration of regions inSection 4.2, we can use the bijection to enumerate vertices of B . For example,the vertex with smallest labelling will always correspond to the boundary of acontractable region. We refer the reader to Figure 5 for an illustration of thislabelling. Coefficients along the blue loops.
Our next goal is to modify β in orderto make all the remaining red edges (i.e. edges in the graph B ) k -divisible. Notethat the 1-chain β might not be an integral 1-cycle, but it is a 1-cycle mod k . Wewill now exploit this property to analyze the behavior of the 1-chain β along theblue loops. Proposition 2.
Let γ be any blue loop, oriented clockwise. Then the coefficientson the blue edges are all congruent to each other modulo k , that is, if ( a, b, c, d ) and ( a (cid:48) , b (cid:48) , c (cid:48) , d (cid:48) ) are the coefficients along any two blue edges in a given blue loop, then ( a, b, c, d ) ≡ ( a (cid:48) , b (cid:48) , c (cid:48) , d (cid:48) ) mod k . Moreover, along any red edge in the graph B , thecoefficient is congruent to zero mod k , except possibly if the edge has an even label,in which case the coefficient is congruent to (0 , , ˆ z, − ˆ z, , . . . , for some z (whichmay vary from edge to edge).Proof. To see this, we argue inductively according to the ordering of the blue con-nected components (see Section 4.5).Base Case. For the initial case, consider the blue loop corresponding to vertex v in the tree B . By hypothesis, this blue loop γ has a single (red) edge incidentto it which is potentially not k -divisible, corresponding to the unique edge in thetree B incident to v . Let w denote the single vertex on γ where that red edgeis incident, allowing us to view γ as a path starting and terminating at w . Sinceall the remaining red edges incident to γ are k -divisible, applying Lemma 4 inSection 4.11 (with all the ˆ z ≡
0) for each incident k -divisible red edge shows thatall the coefficients along the path γ are congruent to each other mod k . Note that,in this base case, we are always in the cases i < j < k or j < k < i of Lemma 4,according to whether the k -divisible red edge lies in the unbounded or boundedregion determined by the blue loop γ . This establishes the first statement of theProposition. To get the second statement, we apply Lemma 5 in Section 4.11 atthe vertex w , and we are done.Inductive Step. Now inductively, let us assume that we are focusing on the blueloop γ i corresponding to some vertex v i in B . We assume that all the blue loops γ j corresponding to vertices v j with j < i already satisfy the desired property. Wealso assume that all red edges in the graph B connected to vertices v j with j < i have coefficients of the form described in the Proposition.From the directed structure of the graph B , the vertex v i has a unique edge e connecting to a vertex v j with index j > i , and all the remaining edges in B connectto a v j for some j < i .By the inductive hypothesis, this tells us that all but one of these red edges havecoefficients congruent to (0 , , ˆ z j , − ˆ z j , , . . . ,
0) for some z j (which might depend onthe edge). Again, viewing γ i as a path starting and terminating at the same vertex w (where e is incident to γ ), we may apply Lemma 4 and conclude that γ i hascoefficients along all edges that are congruent to each other. Applying Lemma 5 atthe vertex w , shows that the coefficients along the edge e must also be of the form(0 , , ˆ z, − ˆ z, , . . . ,
0) for some z . This completes the inductive step and the proof ofthe Proposition. (cid:3) Equivalence classes of red edges.
Now consider a red edge which is po-tentially not k -divisible, corresponding to an edge in the graph B joining vertices QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 15 v i to v j . The red edge thus joins the blue loop γ i to the blue loop γ j . FromProposition 2, we see that the coefficient along the red edge must be congruent to(0 , , ˆ z, − ˆ z, , . . . ,
0) for some z . In particular, there is a single residue class thatdetermines the coefficients along the red edge (modulo k ).Next let us momentarily focus on a blue loop γ , and assume the coefficients alongthe edges of γ are all congruent to ( a, b, c, d ) modulo k , as ensured by Proposition 2(Figure 6). We define an equivalence relation on all the red edges with even labelincident to γ , by defining the two equivalence classes:(a) the incident red edges that lie in the bounded region corresponding to γ , and(b) those that lie in the unbounded region.It follows from Lemma 4 (Section 4.11), that all edges in the equivalence class(a) have corresponding coefficients congruent to (0 , , ˆ z j , − ˆ z j , , . . . ,
0) where each z j ≡ b − a , while all edges in equivalence class (b) have corresponding coefficientscongruent to (0 , , ˆ z j , − ˆ z j , , . . . ,
0) where each z j ≡ c − a (Figure 6). z j ≡ b − aγ congruent to ( a, b, c, d ) z j ≡ c − a Figure 6.
Bad red edges incident to blue loop γ . All interior edges have coefficients satisfying one congruence, while all exterior edges have coefficients satisfying a different congruence (Sec-tion 4.7).This equivalence relation is defined locally, and can be extended over all blueloops in the 1-skeleton, resulting in an equivalence relation on the collection of allred edges with even label. Observe that, by construction, each equivalence class hasthe property that there is a single corresponding residue class z mod k , with theproperty that all the edges within that equivalence class have coefficient congruentto (0 , , ˆ z, − ˆ z, , . . . , z is the same for the entire equivalence class. Corollary 5.
The edges in the graph B that are not k -divisible are a finite unionof equivalence classes for this relation.Proof. Let e be an edge in B , and assume that e is equivalent to an edge e (cid:48) whichis not an edge in B . Since all edges that are not in B are k -divisible, it followsthat the coefficient on e (cid:48) is congruent to zero mod k . Thus the value of z for theequivalence class E containing e (cid:48) is z = 0. Since e ∈ E , this forces e to be k -divisible,a contradiction. (cid:3) Establishing k -divisibility of the remaining red edges. Observe thatthe edges in each equivalence class form a connected subgraph, and hence a subtree(see Lemma 2), of the graph B . This collection of subtrees partitions the graph B .Any vertex of B is incident to at most two such subtrees – the incident red edgeslying “inside” and “outside” the corresponding blue loop.We now proceed to establish k -divisibility of the remaining red edges for ourchain. Fix an equivalence class E of red edges, and associate to it a 1-chain α E whose coefficients are given as follows:(1) if a blue loop γ has an incident red edge e ∈ E , and e lies in the bounded region of γ , then assign (0 , , ,
0) to each blue edge on γ ;(2) if a blue loop γ has an incident red edge e ∈ E , and e lies in the unbounded region of γ , then assign (0 , , ,
0) to each blue edge on γ ;(3) along the red edges in the equivalence class (recall that all these edges haveeven labels), assign ± (0 , , ˆ1 , − ˆ1 , , . . . , B – and thus there are no cycles (these couldhave potentially forced the sign along an edge to be both positive and negative).Another key feature of the 1-chains α E is that they are linearly independent. Moreprecisely, two distinct equivalence classes E , E (cid:48) have associated 1-chains α E and α E (cid:48) whose supports are disjoint, except possibly along a single blue loop γ . In the casewhere the supports overlap along γ , adding multiples of α E does not affect the z -value along the class E (cid:48) (and vice versa).It is now immediate from the equality case of Lemma 4 in Section 4.11 that the1-chain α E is in fact an integral 1-cycle. Subtracting multiples of α E from our givenchain β , we may thus obtain a homologous 1-chain for which all the red edges in E are now k -divisible. Repeating this for each of the equivalence classes, we havenow obtained a homologous 1-chain (still denoted β ) for which all the red edges are k -divisible. Establishing k -divisibility of the remaining blue edges. We now haveobtained a 1-chain with prescribed differential, whose coefficients along all red edgesare k -divisible. It remains to establish k -divisibility of the blue loops for the 1-chain.If γ is one of the blue loops, then since all incident red edges are k -divisible, wesee that all the edges on γ have coefficients which are congruent to either ( a, b, c, d ),( a, b, a, b ), ( a, a, c, c ), or ( a, a, a, a ), according to the equivalence classes that areincident to γ (see also Proposition 2).Let us discuss, as an example, the case ( a, b, a, b ). Note that this case occursif the only incident red edges to γ with even label lie in the unbounded regiondetermined by γ . Consider the pair of integral 1-cycles α , α supported on γ ,obtained by assigning to each edge on γ the coefficient (1 , , ,
0) and (0 , , , α , α are in fact1-cycles. By adding multiples of α , α , we can now arrange for the coefficientsalong the blue loop γ to all be k -divisible. The three other cases can be dealt withsimilarly; we leave the details to the reader.4.10. Completing the proof.
Performing this process described in Section 4.9for all the blue components, we finally obtain the desired 1-chain β (cid:48) . Since β (cid:48) nowsatisfies properties (1) and (2) mentioned at the beginning of the proof, we conclude QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 17 that the given hypothetical torsion class α ∈ H was in fact the zero class. Thiscompletes the proof of Theorem 4. Remark . It is not obvious how to adapt the strategy in the geometric proof aboveto the case when other vertex stabilizer types are allowed. In the case of vertexstabilizers of the type ∆(2 , , , , , , , , Local Analysis.
The geometric proof that H
Fin (Γ; R C ) is torsion free, The-orem 4, relies on a detailed local analysis of the induction homomorphism at thevertices of the polyhedron P . We state and prove the results needed here. Althoughrather technical, they are all, unless an explicit proof is given, straightforward con-sequences of the induction homomorphisms in Appendix B. Let us introduce somenotation. Throughout this section, α will denote an integral 1-chain that is alsoa 1-cycle (mod k ), i.e. ∂ ( α ) ≡
0. Our goal is to understand how this conditionconstrains the coefficients of α .At every vertex v = v ijk , there are three incident edges e = e ij , e = e ik and e = e jk , and let x , x and x be α projected along those edges, writtenas column vectors. Write A for the integer matrix representing the inductionhomomorphism from e to v , that is, R C ( (cid:104) s i , s j (cid:105) ) −→ R C ( (cid:104) s i , s j , s k (cid:105) ) (as always,we will implicitly identify representation rings with free abelian groups, via thebases explicitly described in Appendix A), and define A and A analogously for e and e . So each of these matrices is a submatrix of ∂ in matrix form. Then thevalue of ∂ ( x ) at the vertex v = v ijk (that is, projected to R C ( (cid:104) s i , s j , s k (cid:105) )) is givenby the matrix product ( ± A | ± A | ± A ) · x x x , with signs depending on edgeorientations. The product above is zero modulo k , by the hypothesis on α being a1-cycle mod k . We can reduce modulo k all the entries and, abusing notation, stillcall the resulting matrices and vectors A , A , A , x , x and x . Furthermore, forsimplicity, let us redefine A as − A etc as needed to take account of the chosenorientations. Then we have that the column vector representing α locally at v (i.e. along the incident edges) is in the kernel of the matrix representing ∂ locallyat v , that is, x x x ∈ ker ( A | A | A ) . One important consequence is that we canperform row operations on the matrix A = ( A | A | A ) without changing its kerneland, in particular, we may row reduce A , for instance into its Hermite normal form,to simplify calculations. (Obviously, row reduction must be performed modulo k ,that is, in Z /k Z .) Another consequence is that, to study the consequences ofestablishing k -divisibility of an edge, we only need to remove the correspondingmatrix block and vector. For example, if k -divisibility of e has been establishedfor α , that is, if x is zero modulo k , then the equation above is equivalent to (cid:18) x x (cid:19) ∈ ker ( A | A ) , and we can now row reduce this matrix to help us calculateits kernel, if needed. Recall that, throughout Section 4, we are only interested in the case where all vertices have stabilizers of the form ∆(2 , , m ), with m > , , m ), m >
2, vertex, recall that we assume j < k and we have, fromFigures 12, 13 and 14, induction matrices M (1) < ,m (if i < j ) or M (1) > ,m (if i > j ), M (2) < ,m (if i < k ) or M > ,m (if i > k ), and M m,m where M (1) < ,m = (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98)
01 1 0 0... ... ... ...1 1 0 00 0 1 00 0 0 1 (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98)
00 0 1 1... ... ... ...0 0 1 1 , M (2) < ,m = (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98)
01 1 0 0... ... ... ...1 1 0 00 0 1 00 0 0 1 (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) (cid:98)
10 0 1 1... ... ... ...0 0 1 1 , M m,m = (cid:32) Id c ( D m ) Id c ( D m ) (cid:33) , and M (1) > ,m , respectively M (2) > ,m , equals M (1) < ,m , respectively M (2) < ,m , with the 2ndand 3rd columns interchanged.In the following lemmas, recall that α is an integral 1-chain which is also a 1-cycle mod k . Moreover, in Lemmas 4 and 5, let (cid:98) denote coefficients that onlyappear when m is even, and recall the standard labelling of faces: the m -edge liesbetween the faces labelled F j and F k , and the labelling always satisfies (withoutloss of generality) that j < k . Finally, recall that we refer to an edge with stabiliser D m as an m -edge , or edge of type m . Lemma 3.
Let α be an integral -chain, which we assume is also a -cycle mod k .Let F , F be a pair of adjacent -faces, sharing a common edge e , with endpoint v whose stabilizer is of the form ∆(2 , , m ) , with m > . Assume that we are givenan η = ( n , m ) in the representation ring R C ( C ) associated to the stabilizer ofthe -face F . Then there exists a choice of η = ( n , m ) in the representationring R C ( C ) associated to the stabilizer of the -face F , with the property that α + ∂ ( η + η ) has coefficient along e congruent to zero mod k (i.e. the edge e isnow k -divisible).Proof. The edge e has stabilizer D m , with m >
2. We will assume the orientationsalong the edges and faces are as given in Figure 7. Assume the coefficients of α supported on the edge e are given by ( a, b, ˆ c, ˆ d, r , . . . , r k ). We choose η :=( a + n , r + m − a ). A straightforward computation using the induction formulasshows that, with this choice of η , the coefficient of α (cid:48) := α + ∂ ( η + η ) along theedge e is of the form z := (0 , b (cid:48) , ˆ c (cid:48) , ˆ d (cid:48) , , r (cid:48) , . . . , r (cid:48) k ). That is to say, we chose η inorder to force a (cid:48) = r (cid:48) = 0.We are left with checking that the remaining coefficients of α (cid:48) are all congruentto zero mod k . To see this, we use the fact that α (cid:48) is also a 1-cycle mod k . Fromthe labelings of the faces around the vertex v , and the order in which we label faces(one region at a time), we see that we are in one of the two cases i < j < k or QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 19 e ¯ e ¯ e ¯ e ¯ e F i +1 F j Figure 7.
Local picture near the edge e (Lemma 3). j < k < i . Let us consider the case i < j < k , and assume that the coefficients alongthe 2-edges incident to v are given by x := ( x , x , x , x ) and y := ( y , y , y , y ).As α (cid:48) is a 1-cycle mod k , we have (cid:16) M (1) < ,m (cid:12)(cid:12)(cid:12) − M (2) < ,m (cid:12)(cid:12)(cid:12) − M m,m (cid:17) xyz ≡ , Note that the last k rows of the matrix are identical, giving rise to k identicalrelations x + x − y − y + r (cid:48) i ≡ ≤ i ≤ k ). Since r (cid:48) = 0, these equationsimmediately imply that all the remaining r (cid:48) i ≡ m is odd. The first and third row of the matrix giverise to equations x − y ≡ x + x − y − y ≡
0, forcing x − y ≡ x − y + b (cid:48) ≡
0, which immediately gives b (cid:48) ≡ i < j < k and m is odd. The case where m is evenis analogous – one just uses the equations obtained from the first five rows of thematrix to conclude that a (cid:48) , b (cid:48) , ˆ c (cid:48) , and ˆ d (cid:48) are all congruent to zero mod k .Finally, if j < k < i , then one proceeds in a completely similar manner, but usingthe block matrix (cid:16) M (1) > ,m | − M (2) > ,m | − M m,m (cid:17) instead. It is again straightforwardto work through the equations – we leave the details to the reader. (cid:3) Lemma 4.
Consider a vertex of type ∆(2 , , m ) , m > , with the incident -edges oriented compatibly. If the coefficients of α along the m -edge are congruentto (0 , , ˆ z, − ˆ z, , . . . , for some z , then the coefficients ( a, b, c, d ) and ( a (cid:48) , b (cid:48) , c (cid:48) , d (cid:48) ) along the pair of -edges satisfy the following congruences:(i) if i < j < k , then ( a, b, c, d ) ≡ ( a (cid:48) , b (cid:48) , c (cid:48) , d (cid:48) ) , and ˆ z ≡ b − a ≡ d − c ;(ii) if j < i < k , then ( a, b, c, d ) ≡ ( a (cid:48) , c (cid:48) , b (cid:48) , d (cid:48) ) , and ˆ z ≡ c − a ≡ d − b ;(iii) if j < k < i , then ( a, b, c, d ) ≡ ( a (cid:48) , b (cid:48) , c (cid:48) , d (cid:48) ) , and ˆ z ≡ c − a ≡ d − b ;and we have oriented the m -edge so that the vertex is its source. (With the oppositeorientation, simply replace ˆ z by − ˆ z .) Moreover, the same statement holds if onechanges all congruences to equalities.Proof. Let x = ( a, b, c, d ), x = ( a (cid:48) , b (cid:48) , c (cid:48) , d (cid:48) ) and x = (0 , , ˆ z, − ˆ z, , . . . ,
0) be thecoefficients of α along the edges incident to the vertex. Consider the case i < j < k first. Since α is a 1-cycle mod k , (cid:16) M (1) < ,m (cid:12)(cid:12)(cid:12) − M (2) < ,m (cid:12)(cid:12)(cid:12) − M m,m (cid:17) x x x ≡ , which gives a − a (cid:48) ≡ b − b (cid:48) ≡ b − a (cid:48) − ˆ z ≡ a − b (cid:48) + ˆ z ≡ a + b − a (cid:48) − b (cid:48) ≡ c − c (cid:48) ≡ d − d (cid:48) ≡ d − c (cid:48) − ˆ z ≡ c − d (cid:48) + ˆ z ≡
0, and c + d − c (cid:48) − d (cid:48) ≡ j < i < k and j < k < i ,are analogous but for the block matrix (cid:16) M (1) > ,m | − M (2) < ,m | − M m,m (cid:17) , respectively (cid:16) M (1) > ,m | − M (2) > ,m | − M m,m (cid:17) . The former gives the same congruences but with b and c interchanged, and the latter with b and c , and b (cid:48) and c (cid:48) , interchanged. Forthe opposite orientation of the m -edge, replace − M m,m by M m,m in the calculationabove. (cid:3) Lemma 5.
Consider a vertex of type ∆(2 , , m ) , m > , with the incident -edgesoriented compatibly. Assume the coefficients along the -edges are both congruentto ( a, b, c, d ) , that the m -edge is oriented compatibly with the first 2-edge, and thatthe faces are labelled so that i < j < k or j < k < i (so we are excluding thecase j < i < k ). Then the m -edge coefficients are congruent to (0 , , ˆ z, − ˆ z, , . . . , where(i) if i < j < k , then ˆ z ≡ a − b ;(ii) if j < k < i , then ˆ z ≡ a − c .(If we reverse the orientation on the m -edge, the congruencies above hold with ˆ z replaced by − ˆ z .) In particular, if m is odd, the m -edge is automatically k -divisible.Proof. We are assuming x ≡ x ≡ ( a, b, c, d ), and that x = ( x, y, ˆ z, ˆ t, r , . . . , r k )are the coefficients of α along the edges incident to the vertex. Consider the case i < j < k first. Since α is a 1-cycle mod k , (cid:16) M (1) < ,m (cid:12)(cid:12)(cid:12) − M (2) < ,m (cid:12)(cid:12)(cid:12) − M m,m (cid:17) x x x ≡ , which gives x ≡ a − a ≡ y ≡ b − b ≡
0, ˆ c ≡ a − b , ˆ d ≡ b − a , while all theremaining equations are of the form r i ≡ ( a + b ) − ( a + b ) ≡
0. The claim follows.The case j < k < i is completely analogous, but uses instead the block matrix (cid:16) M (1) > ,m | − M (2) > ,m | − M m,m (cid:17) . The details are left to the reader. (cid:3) It is perhaps worth noting that the analogue of Lemma 5 is false if the facesare enumerated to satisfy j < i < k . In particular, the corresponding block ma-trix (cid:16) M (1) > ,m | − M (2) < ,m | − M m,m (cid:17) leads to, for example, y ≡ b − c , which is notnecessarily zero.5. No torsion in H – the linear algebra approach In this section, we give a proof of Theorem 3 inspired by the representation ringsplitting technique of [23]. We do this by establishing a criterion for H
Fin (Γ; R C ) tobe torsion-free. Our criterion is efficient to check, and only requires elementary lin-ear algebra. Furthermore, we will see it is satisfied for Γ a 3-dimensional hyperbolicCoxeter group. QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 21
The verification of our criterion relies on simultaneous base transformations ofthe representation rings, bringing the induction homomorphisms into the desiredform. For the 3-dimensional hyperbolic Coxeter groups, these transformations arecarried out in Appendices A and B. In Section 6 which comes next, we will alsosee that this condition is satisfied for several additional classes of groups that hadpreviously been considered by other authors.
Definition 1.
The vertex block of a given vertex v in a Bredon chain complexdifferential matrix ∂ consists of all the blocks of ∂ that are representing mapsinduced (on complex representation rings from Γ e → Γ v ) by edges e incident to v .We represent elements in the Bredon chain complex as column vectors. So thematrix D for the differential ∂ is a rank Z C × rank Z C matrix, acting by leftmultiplication on a column vector in C . For a vertex v , denote by n the rank of R C (Γ v ), and by n , n , n , the ranks of the representation rings corresponding tothe three edges e , e and e incident to v . Then the vertex block for v is a submatrixof D of size n × ( n + n + n ). Since vertex blocks have been constructed to containall entries from incident edges, we note that the rest of the entries in their rows arezero. Theorem 5.
If there exists a base transformation such that all minors in all vertexblocks are in the set {− , , } , then H Fin (Γ; R C ) is torsion-free.Proof of Theorem 5. We start by recalling a general result on Smith Normal Forms,already observed by Smith [25]. Denote by d i ( A ) the i -th determinant divisor d i ( A ),defined to be the greatest common divisor of all i × i minors of a matrix A when i ≥
1, and to be d ( A ) := 1 when i = 0. Then the elementary divisors of thematrix A , up to multiplication by a unit, coincide with the ratios α i = d i ( A ) d i − ( A ) .Let us use the notationpre-rank( ∂ ) := rank Z C − rank Z ker ∂ , where C is the module of 1-chains in the Bredon chain complex (Equation (6)).Observe that, if A is any i × i submatrix of D of non-zero determinant, and i < pre-rank( ∂ ), then A can be expanded to some ( i +1) × ( i +1) submatrix of non-zerodeterminant.Now H Fin (Γ; R C ) is torsion-free if and only if α i = ± ≤ i ≤ pre-rank( ∂ ).From the discussion above, it is sufficient to find, for each 1 ≤ i ≤ pre-rank( ∂ ), an i × i minor in the Bredon chain complex differential matrix ∂ with determinant ±
1. We produce such a minor by induction on i .Base Case. For i = 1, we observe there are vertices with adjacent edges, hence thereare non-zero vertex blocks. As by assumption all the entries in the vertex blocksare in the set {− , , } , there exists an entry of value ± ≤ i ≤ pre-rank( ∂ ), and assume we already have an ( i − × ( i −
1) minor of ∂ of value ±
1, corresponding to a submatrix B . We want to findan i × i minor of ∂ of value ± V , choose a maximal square submatrix B ◦ of B whichis disjoint from the rows and columns of V . At the two extremes, this submatrixcould be empty (if B is contained in V ) or could coincide with B (if B is completely disjoint from V ). Note that, after possibly permuting rows, we get a square sub-block M of V such that the submatrix B takes the form B = det (cid:18) M ∗ B ◦ (cid:19) . Then in particular ± B ) = det( M ) · det( B ◦ ), which forces det( B ◦ ) = ± B ◦ to an i × i block B (cid:48) by picking a submatrix M (cid:48) inside V of size i − size( B ◦ ). Such an extension might not be possible, but when itis, the resulting block B (cid:48) takes the form (possibly after permuting rows) B (cid:48) = det (cid:18) M (cid:48) ∗ B ◦ (cid:19) . Consider the collection of all i × i blocks obtained in this manner, and note thatfor any such block, we have det( B (cid:48) ) = det( M (cid:48) ) · det( B ◦ ) = ± i ≤ pre-rank( ∂ ), there exists a vertex block V for which this constructionyields an i × i -block with det( B (cid:48) ) (cid:54) = 0. We have that M (cid:48) is a minor in the vertexblock V , so by hypothesis det( M (cid:48) ) ∈ {− , , } . Since det( B (cid:48) ) (cid:54) = 0, we concludethat det( M (cid:48) ) = ±
1. And as we already noted above, the submatrix B ◦ of B (cid:48) satisfies det( B ◦ ) ∈ {− , } . This implies our submatrix B (cid:48) satisfies det( B (cid:48) ) =det( M (cid:48) ) · det( B ◦ ) = ±
1, which completes the inductive step and hence the proof ofthe theorem. (cid:3)
Our proof of Theorem 3 now reduces to verifying the hypotheses of Theorem 5,when Γ is a 3-dimensional hyperbolic reflection group. We will rely on the simul-taneous base transformations that can be found in Appendix A.
Proposition 3.
For a system of finite subgroups of types A × C , S , S × C , ∆(2 , ,
2) = ( C ) and ∆(2 , , m ) = C × D m for m ≥ as vertex stabilizers, withtheir three 2-generator Coxeter subgroups as adjacent edge stabilizers, there is asimultaneous base transformation such that all vertex blocks have all their minorscontained in the set {− , , } .Proof. We apply the base transformation specified in Appendix A. Then we havethat all of the induced maps have all of their entries in the set {− , , } (seeAppendix B, all tables referenced in this proof can be found there). Next, for eachvertex stabilizer type, we assemble the vertex blocks from the three vertex-edge-adjacency induced maps.Let us provide full details for the case of vertex stabilizer ∆(2 , , m ) = C × D m for m ≥
3. By Tables 29 and 30, the vertex block of a stabilizer of type C × D m for m ≥ ± and one block ± (cid:18) identity matrix of size m +32 (cid:19) . QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 23
Note that all the columns in this matrix have a very special form: all but one of theentries are zero, and the single non-zero entry is ±
1. An easy induction shows that,when checking whether the minors all take value in the set { , ± } , such columnscan always be discarded (and likewise for rows). This fact is very useful for reducingthe size of the matrices to check. For the matrix above, this fact immediately letsus conclude that all minors are in { , ± } .For m ≥ D m × C D m (cid:44) → D m × C D (cid:44) → D m × C D (cid:44) → D m × C ρ ⊗ χ ↓ ρ ⊗ ( χ − χ ) ↓ ρ ⊗ ( χ − χ ) ↓ − ρ ⊗ ( χ − χ ) ↓ ρ ⊗ ( φ − χ − χ ) ↓ ρ ⊗ ( φ p − φ p − ) ↓ ... 0 0 0 0 0 ... 0 ... ... ... ... ... ... ... ... ρ ⊗ ( φ m − − φ m − ) ↓ ρ − ρ ) ⊗ χ ↓ ρ − ρ ) ⊗ ( χ − χ ) ↓ ρ − ρ ) ⊗ ( χ − χ ) ↓ ρ − ρ ) ⊗ ( χ + χ − χ − χ ) ↓ ρ − ρ ) ⊗ ( φ − χ − χ ) ↓ ρ ⊗ ( φ p − φ p − ) ↓ ... ... ... ... ... ... ... ... ... ρ ⊗ ( φ m − − φ m − ) ↓ Again, we can discard all the rows and columns which have at most one entry ± ± ±
10 1 − , for which we can easily check that all minors lie in { , ± } .For m ≥ D m × C D m (cid:44) → D m × C D (cid:44) → D m × C D (cid:44) → D m × C ρ ⊗ χ ↓ ρ ⊗ ( χ − χ ) ↓ ρ ⊗ ( χ − χ ) ↓ ρ ⊗ ( χ − χ ) ↓ − ρ ⊗ ( φ − χ − χ ) ↓ ρ ⊗ ( φ p − φ p − ) ↓ ... 0 0 0 0 0 ... 0 ... ... ... ... ... ... ... ...( φ m − − φ m − ) ↓ ρ − ρ ) ⊗ χ ↓ ρ − ρ ) ⊗ ( χ − χ ) ↓ ρ − ρ ) ⊗ ( χ − χ ) ↓ ρ − ρ ) ⊗ ( χ + χ − χ − χ ) ↓ ρ − ρ ) ⊗ ( φ − χ − χ ) ↓ ρ ⊗ ( φ p − φ p − ) ↓ ... ... ... ... ... ... ... ... ... ρ ⊗ ( φ m − − φ m − ) ↓ Again, we can discard the rows and columns which have at most one entry ± ± ± ± ±
10 1 1 0 0 0 0 0 00 − , for which we can easily check that it has all its minors in { , ± } . This completesthe verification of the vertex block condition in the case of vertices with stabilizer∆(2 , , m ) = C × D m for m ≥ http://math.uni.lu/˜rahm/vertexBlocks/ . Note that for the groupsunder consideration, the matrix rank of the vertex block is at most 7, so the 8 × n × n -minors for n ≤
7. Thiscomputer check verifies the minor condition for the vertex blocks associated to allremaining vertex stabilizers, and completes the proof of the theorem. (cid:3)
Corollary 6.
For any Coxeter group Γ having a system of finite subgroups of types ∆(2 , ,
2) = ( C ) , ∆(2 , , m ) = C × D m for m ≥ , S , S × C or A × C asvertex stabilizers, we have that the Bredon homology group H Fin (Γ; R C ) is torsion-free.Remark . (a) When trying to extend the proof of Theorem 5 to H Fin n (Γ; R C ) for n >
0, one should take into account the natural map H
Fin n (Γ; R C ) → H n (BΓ; Z )described by Mislin [21], which is an isomorphism for n > dim EΓ sing + 1,where EΓ sing consists of the non-trivially stabilized points in EΓ. Hence suchan extension of the theorem can only be useful when n ≤ dim EΓ sing + 1.(b) Note that the search for suitable base transformations for a given group Γ(as described in Appendix A in our case), can be quite laborious. If the readerwants to apply Theorem 5 for a given group Γ, it is prudent to first construct thevertex blocks without any base transformation and compute their elementarydivisors. If there exists a suitable simultaneous base transformation whichsatisfies the hypotheses of Theorem 5, then those elementary divisors must bein the set {− , , } .6. Further examples with torsion-free H Fin (Γ; R C )In this section we briefly steer away from Coxeter groups, and instead givesome further examples illustrating our criterion for the Bredon homology groupH Fin (Γ; R C ) to be torsion-free.6.1. The Heisenberg semidirect product group.
Let us show that H
Fin (Γ; R C )is torsion-free for Γ the Heisenberg semidirect product group of L¨uck’s paper. InTables 3 and 4, we transform the character tables of all the non-trivial finite sub-groups of the Heisenberg semidirect product group, as identified by L¨uck [15]. QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 25 C e sρ ρ − (cid:55)→ C e sρ + ρ ρ − Table 3.
Character table of the cyclic group C of order 2, withgenerator s . C e s s s ρ ρ − − ρ i − − iρ − i − i (cid:55)→ C e s s s ρ ρ − ρ − − ρ − ρ i − − − i − ρ − ρ − i i Table 4.
Character table of the cyclic group C of order 4, withgenerator s . We let i = − R C ( H ) → R C ( G ) appearing in any possible Bredon chain complex. C (cid:44) → C e s ( ·| ρ + ρ ) ( ·| ρ ) ρ ↓ ρ − ρ ) ↓ ρ − ρ ) ↓ − ρ + ρ − ρ − ρ ) ↓ Table 5.
The only non-trivial inclusion C (cid:44) → C of a cyclic groupof order 2 into a cyclic group of order 4: s (cid:55)→ s .C (cid:44) → C e e ( ·| ρ + ρ ) ( ·| ρ ) ρ ↓ ρ − ρ ) ↓ ρ − ρ ) ↓ ρ + ρ − ρ − ρ ) ↓ Table 6.
The trivial inclusion C (cid:44) → C of a cyclic group of or-der 2 into a cyclic group of order 4: s (cid:55)→ e. Obviously, any concatenation of copies of the three matrices given in Tables 5,6 and 7 yields a matrix with all of its minors contained in the set {− , , } . Forthe inclusions into cyclic groups of order 2, an analogous (and even simpler) proce-dure works. Hence by Theorem 5, H Fin (Γ; R C ) is torsion-free for Γ the Heisenbergsemidirect product group of L¨uck’s article [15] . Crystallographic groups.
Davis and L¨uck [5] consider the semidirect prod-uct of Z n with the cyclic p -group Z /p , where the action of Z /p on Z n is given C (cid:44) → C e ( ·| τ ) ρ ↓ ρ − ρ ) ↓ ρ − ρ ) ↓ ρ + ρ − ρ − ρ ) ↓ Table 7.
The only inclusion { } (cid:44) → C of the trivial group into acyclic group of order 4.by an integral representation, which is assumed to act freely on the complementof zero. The action of this semidirect product group Γ on EΓ ∼ = R n is crystallo-graphic, with Z n acting by lattice translations, and Z /p acting with a single fixedpoint. In particular, all cell stabilizers are trivial except for one orbit of vertices ofstabilizer type Z /p . So all maps in the Bredon chain complex are induced by thetrivial representation, and we can easily apply Theorem 5 to see that H Fin (Γ; R C )is torsion-free for Γ.7. cf (Γ) and χ ( C ) from the geometry of P Let Γ be the reflection group of the compact 3-dimensional hyperbolic poly-hedron P . In this final section, we compute the number of conjugacy classes ofelements of finite order of Γ, cf (Γ), and the Euler characteristic of the Bredonchain complex (5), χ ( C ), from the geometry of the polyhedron P . This gives usexplicit combinatorial formulas for the Bredon homology and equivariant K -theorygroups computed in our Main Theorem.7.1. Conjugacy classes of elements of finite order.
We now give an algorithmto calculate cf (Γ), the number of conjugacy classes of elements of finite order inthe Coxeter group Γ. We know that each element of finite order can be conjugatedto one which stabilizes one of the k -dimensional faces of the polyhedron, for some k ∈ { , , } . Of course, the only element which stabilizes all faces is the identityelement. Let us set that aside, and consider the non-identity elements, to which weassociate the integer k . We now count the elements according to the integer k , indescending order.Case k = 2: These are the conjugacy classes represented by the canonical generatorsof the Coxeter group Γ. The number of these is given by the total number |P (2) | of facets of the polyhedron P .Case k = 1: These elements are edge stabilizers which are not conjugate to the sta-bilizer of a face. We first note that there are some possible conjugacies between edgestabilizers. Geometrically, these occur when there is a geodesic γ ⊂ H whose pro-jection onto the fundamental domain P covers multiple edges inside the 1-skeleton P (1) . A detailed analysis of when this can happen is given in [11]. Following thedescription in that paper, we decompose the 1-skeleton into equivalence classes ofedges, where two edges are equivalent if there exists a geodesic whose projectionpasses through both edges. Denote by [ P (1) ] the set of equivalence classes of edges,and note that each equivalence class [ e ] has a well defined group associated to it,which is just the dihedral group Γ e stabilizing a representative edge. We can thuscount the conjugacy classes in the corresponding dihedral group, and discard the QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 27 three conjugacy classes already accounted for (the conjugacy class of the two canon-ical generators counted in case k = 2, as well as the identity). Thus the contributionfrom finite elements of this type is given by (cid:88) [ e ] ∈ [ P (1) ] ( c (Γ e ) − . (Recall that c ( D m ), the number of conjugacy classes in a dihedral group of order2 m , is m/ m even, and ( m − / m is odd.)Case k = 0: Finally, we consider the contribution from the elements in the vertexstabilizers which have not already been counted . That is to say, for each vertex v ∈P (0) , we count the conjugacy classes of elements in the corresponding 3-generatedspherical triangle group, which cannot be conjugated into one of the canonical 2-generated special subgroups. This number, ¯ c (Γ v ), depends only on the isomorphismtype of the spherical triangle group Γ v , see Table 8. The contribution from thesetypes of finite elements is thus (cid:88) v ∈P (0) ¯ c (Γ v ) . Γ v c (Γ v ) ¯ c (Γ v )∆(2 , , m ) 2 c ( D m ) c ( D m ) − , ,
3) 5 1∆(2 , ,
4) 10 3∆(2 , ,
5) 10 5
Table 8.
Number of conjugacy classes in spherical trianglegroups. The left column is the total number (cf. Appendix A),and the right column the number of those not conjugated into oneof the three canonical 2-generated special subgroups.Combining all these, we obtain the desired combinatorial formula for the numberof conjugacy classes of elements of finite order inside the group Γ: cf (Γ) = 1 + |P (2) | + (cid:88) [ e ] ∈ [ P (1) ] ( c (Γ e ) −
3) + (cid:88) v ∈P (0) ¯ c (Γ v ) . Euler characteristic.
The Euler characteristic of the Bredon chain complexcan be easily calculated from the stabilizers of the various faces of the polyhedron P , according to the formula: χ ( C ) = (cid:88) f ∈P ( − dim( f ) dim( R C (Γ f )) . Depending on the dimension of the faces, we know exactly what the dimension of thecomplex representation ring is (the number of conjugacy classes in the stabilizer): • for the 3-dimensional face (the interior), the stabilizer is trivial, so there isa 1-dimensional complex representation ring; • for the 2-dimensional faces, the stabilizer are Z , and there is a 2-dimensionalcomplex representation ring; • for the 1-dimensional faces e , the stabilizers are dihedral groups, and thereis a c (Γ e )-dimensional complex representation ring; • for the 0-dimensional faces v , the stabilizers are spherical triangle groups,and there is a c (Γ v )-dimensional complex representation ring.Putting these together, we obtain χ ( C ) = − |P (2) | − (cid:88) e ∈P (1) c (Γ e ) + (cid:88) v ∈P (0) c (Γ v ) . Finally, we obtain the desired explicit version of the Main Theorem, expressing the K -theory groups in terms of the geometry of the polyhedron P . Main Theorem ( explicit). Let Γ be a cocompact -dimensional hyperbolic reflec-tion group, generated by reflections in the side of a hyperbolic polyhedron P ⊂ H .Then K ( C ∗ r (Γ)) is a torsion-free abelian group of rank cf (Γ) = 1 + |P (2) | + (cid:88) [ e ] ∈ [ P (1) ] ( c (Γ e ) −
3) + (cid:88) v ∈P (0) ¯ c (Γ v ) , and K ( C ∗ r (Γ)) is a torsion-free abelian group of rank cf (Γ) − χ ( C ) = 2 − |P (2) | + (cid:88) [ e ] ∈ [ P (1) ] ( c (Γ e ) −
3) + (cid:88) e ∈P (1) c (Γ e ) − (cid:88) v ∈P (0) ( c (Γ v ) − ¯ c (Γ v )) , where the values for the c (Γ v ) and ¯ c (Γ v ) are listed in Table 8. References [1] A. Adem. Characters and K -theory of discrete groups. Invent. Math. , 114(3):489–514, 1993.[2] E. M. Andreev. Convex polyhedra of finite volume in Lobaˇcevski˘ı space.
Mat. Sb. (N.S.) , 83(125):256–260, 1970.[3] M. Bo˙zejko, T. Januszkiewicz, and R. J. Spatzier. Infinite Coxeter groups do not have Kazh-dan’s property.
J. Operator Theory , 19(1):63–67, 1988.[4] M. .W. Davis.
The geometry and topology of Coxeter groups . London Mathematical SocietyMonographs Series, 32. Princeton University Press, Princeton, NJ, 2008. xvi+584 pp. ISBN:978-0-691-13138-2; 0-691-13138-4.[5] J. F. Davis and W. L¨uck. The topological K-theory of certain crystallographic groups.
J.Noncommut. Geom. , 7(2):373–431, 2013.[6] R. Flores, S. Pooya, and A. Valette. K-homology and K-theory for the lamplighter groups offinite groups. To appear in
Proc. LMS , arXiv:1610.02798, 2016.[7] N. Higson and G. Kasparov. Operator K -theory for groups which act properly and isometri-cally on Hilbert space. Electron. Res. Announc. Amer. Math. Soc. , 3:131–142, 1997.[8] O. Isely.
K-theory and K-homology for semi-direct products of Z by Z . 2011. Th`ese dedoctorat : Universit´e de Neuchˆatel (Switzerland).[9] G. James amd M. Liebeck Representations and characters of groups . 2nd ed. (English), Zbl0981.20004. Cambridge: Cambridge University Press. viii, 458 p., 2001.[10] J.-F. Lafont, B. A. Magurn, and I. J. Ortiz. Lower algebraic K -theory of certain reflectiongroups. Math. Proc. Cambridge Philos. Soc. , 148(2):193–226, 2010.[11] J.-F. Lafont and I. J. Ortiz. Lower algebraic K -theory of hyperbolic 3-simplex reflectiongroups. Comment. Math. Helv. , 84(2):297–337, 2009.[12] J.-F. Lafont, I. J. Ortiz, and R. J. S´anchez-Garc´ıa. Rational equivariant K -homology of lowdimensional groups. In Topics in noncommutative geometry , volume 16 of
Clay Math. Proc. ,pages 131–163. Amer. Math. Soc., Providence, RI, 2012.[13] M. Langer and W. L¨uck. Topological K -theory of the group C ∗ -algebra of a semi-directproduct Z n (cid:111) Z /m for a free conjugation action. J. Topol. Anal. , 4(2):121–172, 2012.[14] W. L¨uck. Chern characters for proper equivariant homology theories and applications to K -and L -theory. J. Reine Angew. Math. , 543:193–234, 2002.
QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 29 [15] W. L¨uck. K - and L -theory of the semi-direct product of the discrete 3-dimensional Heisenberggroup by Z / Geom. Topol. , 9:1639–1676, 2005.[16] W. L¨uck. Rational computations of the topological K -theory of classifying spaces of discretegroups. J. Reine Angew. Math. , 611:163–187, 2007.[17] W. L¨uck and B. Oliver. Chern characters for the equivariant K -theory of proper G -CW-complexes. In Cohomological methods in homotopy theory (Bellaterra, 1998) , volume 196 of
Progr. Math. , pages 217–247. Birkh¨auser, Basel, 2001.[18] W. L¨uck and H. Reich. The Baum-Connes and the Farrell-Jones conjectures in K - and L -theory. In Handbook of K -theory. Vol. 1, 2 , pages 703–842. Springer, Berlin, 2005.[19] W. L¨uck and R. Stamm. Computations of K - and L -theory of cocompact planar groups. K -Theory , 21(3):249–292, 2000.[20] G. Mislin. Equivariant K -homology of the classifying space for proper actions. In Propergroup actions and the Baum-Connes conjecture , Adv. Courses Math. CRM Barcelona, pages1–78. Birkh¨auser, Basel, 2003.[21] G. Mislin and A. Valette.
Proper group actions and the Baum-Connes conjecture . AdvancedCourses in Mathematics. CRM Barcelona. Birkh¨auser Verlag, Basel, 2003.[22] S. Pooya and A. Valette. K -theory for the C ∗ -algebras of the solvable Baumslag-Solitargroups. Preprint, arXiv:1610.02798, 2016.[23] A. D. Rahm. Sur la K -homologie ´equivariante de PSL sur les entiers quadratiques imagi-naires. Ann. Inst. Fourier , 66(4):1667–1689, 2016.[24] J.-P. Serre.
Linear representations of finite groups . Springer-Verlag, New York-Heidelberg,1977. Translated from the second French edition by Leonard L. Scott, Graduate Texts inMathematics, Vol. 42.[25] H. J. S. Smith. On systems of linear indeterminate equations and congruences.
Philosophicaltransactions of the Royal Society of London , 151:293–326, 1861.[26] M. Yang.
Crossed products by finite groups acting on low dimensional complexes and applica-tions . ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.) – The University of Saskatchewan.
Appendix A. Character tables and base transformations
In this Appendix, we list the character tables of all the groups involved in theBredon chain complex (6), that is, the finite Coxeter subgroups of Γ up to rankthree. These are based on the representation theory described in [9], where all thesecharacter tables are constructed. In the character tables below, rows correspondto irreducible representations, and columns to representatives of conjugacy classes,written in term of the Coxeter generators s , . . . , s n in a fixed Coxeter presentationof Γ, as in (1).In addition, for each character table, we apply elementary row operations toobtain the transformed tables needed for Appendix B, which are in turn used inour proof that H is torsion-free (Section 5). Although the rows of the transformedtables are not irreducible characters, it is easy to check that they still constitutebases for the complex representation rings.Note that, for consistency across subgroups, we will pay particular attention tothe order of the Coxeter generators within a subgroup. We write e for the identityelement in Γ.A.1. Rank 0.
This is the trivial group, with character table given below. eτ Table 9.
Character table of the trivial group.
A.2.
Rank 1.
A rank 1 Coxeter group is a cyclic group of order 2. Write s i for itsCoxeter generator, then its character table is Table 1.A.3. Rank 2.
A finite rank 2 Coxeter group with Coxeter generators s i and s j isa dihedral group of order m = m ij ≥ D m = (cid:104) s i , s j | s i = s j = ( s i s j ) m (cid:105) . The character table of this group is given in Table 2. In order to be consistent,we assume the Coxeter generators are ordered so that i < j . If j < i , then thecharacter table is identical except that the third and fourth rows (the characters (cid:98) χ and (cid:99) χ ) are interchanged, since ( s j s i ) r = ( s i s j ) − r and s i ( s j s i ) r = s j ( s i s j ) − r − .For the case m = 2, that is D = C × C , we will sometimes use the notationcoming from the character table of C (Table 1) instead. (Recall that the irreduciblecharacters of a direct product G × H are obtained from the irreducible charactersof G and H as ρ i ⊗ τ j , where ( ρ i ⊗ τ j ) ( g, h ) = ρ i ( g ) · τ j ( h ).) This gives the notationand characters in Table 10, which are equivalent to Table 2 with ρ ⊗ ρ = χ , ρ ⊗ ρ = χ , ρ ⊗ ρ = χ and ρ ⊗ ρ = χ . As before, we assume i < j , or, if j < i , the second and third rows (characters) must be interchanged. C × C e s i s j s i s j ρ ⊗ ρ ρ ⊗ ρ − − ρ ⊗ ρ − − ρ ⊗ ρ − − Table 10.
Alternative character table of (cid:104) s i , s j (cid:105) ∼ = D = C × C , i < j .We now give the base transformations of the character table of D m needed later,shown in Tables 11, 12 and 13. D e s i s j s i s j (cid:80) χ i χ + χ − χ − − χ + χ − Table 11.
Base transformation of the character table of D .A.4. Rank 3.
A finite rank 3 Coxeter subgroup is one of the spherical trianglegroups ∆(2 , , m ), with m ≥
2, ∆(2 , , , , , , m ij , with the conventions: no edge if m ij = 2, and no label if m ij = 3. QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 31 D m e s i s i s j ( s i s j ) r ( s i s j ) m − χ + χ + 2 (cid:80) m − p =1 φ p m . . . χ + (cid:80) m − p =1 φ p m − . . . (cid:80) m − p =1 φ p m − − . . . − (cid:80) m − p = k φ p ... m − k + 1 ... a k, a k,r a k, m − φ m − b b r b m − Table 12.
Base transformation of the character table of D m , m ≥ a k,r := (cid:80) m − p = k πprm ), b r := 2 cos( π ( m − rm ), and1 < k, r < m − . D m e s i s i s j ( s i s j ) r ( s i s j ) m s j s i s j (cid:80) p =1 χ p + 2 (cid:80) m − p =1 φ p m . . . χ + χ + (cid:80) m − p =1 φ p m . . . − χ + (cid:80) m − p =1 φ p m − − . . . − − χ + χ + (cid:80) m − p =1 φ p m − . . . (cid:80) m − p =1 φ p m − − − ( − r − − ( − m (cid:80) m − p = k φ p ... m − k ... a k, a k,r (cid:80) m − p = k ( − p φ m − b b r − m − Table 13.
Base transformation of the character table of D m , m ≥ a k,r := (cid:80) m − p = k πprm ), b r := 2 cos( π ( m − rm ),1 < k < m − < r < m . m Figure 8.
Coxeter diagrams of rank 3 spherical Coxeter groups.A.4.1. ∆(2 , , . This triangle group is isomorphic to C × C × C and we haveirreducible characters ρ abc := ρ a ⊗ ρ b ⊗ ρ c , a, b, c ∈ { , } , from Table 1, listedin Table 14 below, where ρ a ⊗ ρ b ⊗ ρ c ( x ) = ρ a ( x ) · ρ b ( x ) · ρ c ( x ) for all x =( x , x , x ) ∈ C × C × C . Here we assume that we have ordered the Coxetergenerators s i , s j , s k so that i < j < k . Finally, the base transformation of thecharacter table of ∆(2 , ,
2) needed later is shown in Table 15.A.4.2. ∆(2 , , m ) with m > . This group is isomorphic to C × D m , and hasCoxeter presentation∆(2 , , m ) = (cid:10) s i , s j , s k | s i , s j , s k , ( s i s j ) , ( s i s k ) , ( s j s k ) m (cid:11) , where we have sorted the Coxeter generators s j and s k such that j < k (thegenerator s i is uniquely determined from the presentation). As a direct product C × C × C e s i s j s k s i s j s i s k s j s k s i s j s k ρ := ρ ⊗ ρ ⊗ ρ ρ := ρ ⊗ ρ ⊗ ρ − − − − ρ := ρ ⊗ ρ ⊗ ρ − − − − ρ := ρ ⊗ ρ ⊗ ρ − − − − ρ := ρ ⊗ ρ ⊗ ρ − − − − ρ := ρ ⊗ ρ ⊗ ρ − − − − ρ := ρ ⊗ ρ ⊗ ρ − − − − ρ := ρ ⊗ ρ ⊗ ρ − − − − Table 14.
Character table of (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
2) = C × C × C , i < j < k , from Table 1. C × C × C e s i s j s k s i s j s i s k s j s k s i s j s k ρ ρ − ρ − − − − ρ − ρ − − − − ρ − ρ − − ρ − ρ − − − − ρ − ρ − − ρ − ρ − − ρ − ρ − − Table 15.
Base transformation of the character table of (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
2) = C × C × C , i < j < k .of two groups, the character table of this group can be obtained from those of C (Table 1) and D m (Table 2). This is shown on Table 16, where T D m is the matrixof entries of the character table of D m (Table 2). As explained before, if k < j oneneeds to swap the characters χ and χ , that is, swap ρ ⊗ χ and ρ ⊗ χ , and ρ ⊗ χ and ρ ⊗ χ .The corresponding base transformations for ∆(2 , , m ) ∼ = C × D m are given inTable 17 ( m odd), Table 18 ( m ≥ m ≥ , , . This triangle group has Coxeter presentation(10) ∆(2 , ,
3) = (cid:10) s i , s j , s k | s i , s j , s k , ( s i s j ) , ( s i s k ) , ( s j s k ) (cid:11) . This group is, in standard notation, A , and it is isomorphic to the symmetricgroup S , with Coxeter generators s i = (1 2), s j = (2 3) and s k = (3 4), forinstance. Then we have conjugacy classes (cycle types in S ) represented, in termsof the Coxeter generators, by s i = (1 2), s i s j = (1 3 2), s i s j s k = (1 4 3 2) and s i s k = (1 2)(3 4). In particular, we have the character table for ∆(2 , ,
3) shown inTable 20 below. In this case, the Coxeter generators are unique up to conjugationsince Out( S ) is trivial, hence the choice of s i and s k does not affect the charactertable. QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 33 ∆(2 , , m ) ( s j s k ) r s k ( s j s k ) r s i ( s j s k ) r s i s k ( s j s k ) r ρ ⊗ χ ρ ⊗ χ ρ ⊗ (cid:99) χ T D m T D m ρ ⊗ (cid:99) χ ρ ⊗ φ p ρ ⊗ χ ρ ⊗ χ ρ ⊗ (cid:99) χ T D m − T D m ρ ⊗ (cid:99) χ ρ ⊗ φ p Table 16.
Character table of ∆(2 , , m ), m = m jk > j < k . D m × C e s i ( s i s j ) r α αs i α ( s i s j ) r ρ ⊗ χ ρ ⊗ ( χ − χ ) 0 − − ρ ⊗ ( φ − χ − χ ) 0 0 b r b r ρ ⊗ ( φ p − φ p − ) , a p,r a p,r ( ρ − ρ ) ⊗ χ − − − ρ − ρ ) ⊗ ( χ − χ ) 0 0 0 0 4 0( ρ − ρ ) ⊗ ( φ − χ − χ ) 0 0 0 0 − − b r ( ρ − ρ ) ⊗ ( φ p − φ p − ) , − a p,r Table 17.
Base transformation of the character table of D m × C for m ≥ a p,r := 2 cos( πprm ) − π ( p − rm ), b r :=2 cos( πrm ) −
2, 2 ≤ p ≤ m − and 1 ≤ r ≤ m − , , . This triangle group has Coxeter presentation(11) ∆(2 , ,
4) = (cid:10) s i , s j , s k | s i , s j , s k , ( s i s j ) , ( s i s k ) , ( s j s k ) (cid:11) , and the Coxeter generators are uniquely determined from the presentation. Instandard notation, this is the finite Coxeter group B (or C ). This group isisomorphic to S × C with Coxeter generators, for instance, s i = (1 2) α , s j = (1 3) α and s k = (1 2)(3 4) α , where α is the generator of the C factor. We can chooserepresentatives of the conjugacy classes in terms of the Coxeter generators, for D m × C s i ( s i s j ) r s j s i s j αs i α ( s i s j ) r αs j s i s j ρ ⊗ χ ρ ⊗ ( χ − χ ) − − − − ρ ⊗ ( χ − χ ) 2 c r c r ρ ⊗ ( χ − χ ) − c r − c r ρ ⊗ ( φ − χ − χ ) − b r − b r ρ ⊗ ( φ p − φ p − ) ... 0 a p,r a p,r ρ − ρ ) ⊗ χ − − − ρ − ρ ) ⊗ ( χ − χ ) 0 0 0 4 0 4( ρ − ρ ) ⊗ ( χ − χ ) 0 0 0 − − c r ρ − ρ ) ⊗ ( χ + χ − χ − χ ) 0 0 0 0 − c r ρ − ρ ) ⊗ ( φ − χ − χ ) 0 0 0 0 − b r + c r ) 0( ρ − ρ ) ⊗ ( φ p − φ p − ) 0 0 0 0 − a p,r Table 18.
Base transformation of the character ta-ble of D m × C for m ≥ a p,r := 2 cos( πprm ) − π ( p − rm ), b r := 2 cos( πrm ) − ( − r − c r = ( − r − < p, r < m . D m × C s i ( s i s j ) r s j s i s j αs i α ( s i s j ) r αs j s i s j ρ ⊗ χ ρ ⊗ ( χ − χ ) − − − − ρ ⊗ ( χ − χ ) 0 c r − c r − ρ ⊗ ( χ − χ ) 0 c r c r ρ ⊗ ( φ − χ − χ ) 0 b r b r ρ ⊗ ( φ p − φ p − ) ... 0 a p,r a p,r ρ − ρ ) ⊗ χ − − − ρ − ρ ) ⊗ ( χ − χ ) 0 0 0 4 0 4( ρ − ρ ) ⊗ ( χ − χ ) 0 0 0 0 − c r ρ − ρ ) ⊗ ( χ + χ − χ − χ ) 0 0 0 0 − c r ρ − ρ ) ⊗ ( φ − χ − χ ) 0 0 0 0 − b r ρ − ρ ) ⊗ ( φ p − φ p − ) 0 0 0 0 − a p,r Table 19.
Base transformation of the character tableof D m × C for m ≥ a p,r := 2 cos( πprm ) − π ( p − rm ), b r := 2 cos( πrm ) − c r = ( − r − < p, r < m .example, e ∼ e α ∼ ( s i s j s k ) (1 2) ∼ s i s k (1 2) α ∼ s i (1 2 3) ∼ s i s j (1 2 3) α ∼ s i s j s k (1 2 3 4) ∼ s j s k (1 2 3 4) α ∼ s i ( s j s k ) (1 2)(3 4) ∼ ( s j s k ) (1 2)(3 4) α ∼ s k QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 35 ∆(2 , , e s i s i s j s i s j s k s i s k ξ ξ − − ξ − ξ − − ξ − − Table 20.
Character table of (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , , ∼ = S . S e s i s i s j s i s j s k s i s k ξ (cid:101) ξ := ξ − ξ − − (cid:101) ξ := ξ − ξ − ξ − (cid:101) ξ := ξ − ξ − ξ − − (cid:101) ξ := ξ − ξ − ξ + ξ Table 21.
Base transformation of the character table of∆(2 , , ∼ = S .where ‘ ∼ ’ means ‘conjugate’. Using this representatives, and the fact that ∆(2 , , T S is the matrix of coefficients of the character table of S (Table 20). S × C e s i s k s i s j s j s k ( s j s k ) ( s i s j s k ) s i s i s j s k s i ( s j s k ) s k ρ ⊗ ξ ρ ⊗ ξ ρ ⊗ ξ T S T S ρ ⊗ ξ ρ ⊗ ξ ρ ⊗ ξ ρ ⊗ ξ ρ ⊗ ξ T S − T S ρ ⊗ ξ ρ ⊗ ξ Table 22.
Character table of (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
4) = S × C .The base transformation needed is given in Table 23, where { (cid:101) ξ i } is the trans-formed basis of the character table of S , Table 21.A.4.5. ∆(2 , , . This group has Coxeter presentation(12) ∆(2 , ,
5) = (cid:10) s i , s j , s k | s i , s j , s k , ( s i s j ) , ( s i s k ) , ( s j s k ) (cid:11) S × C (1) (12) (123) (1234) (12)(34) α α (12) α (123) α (1234) α (12)(34) α α − − − − α − − α − − − α α − − − α − α α − − α − α := ρ ⊗ ξ , α := ρ ⊗ (cid:101) ξ , α := ρ ⊗ (cid:101) ξ , α := ρ ⊗ ( (cid:101) ξ + 2 (cid:101) ξ ) − ( ρ ⊗ ξ − ρ ⊗ ξ ), α := ρ ⊗ (cid:101) ξ , α := ρ ⊗ ξ − ρ ⊗ ξ , α := ρ ⊗ ξ − ρ ⊗ ξ + ρ ⊗ ( ξ − ξ ), α := ( ρ − ρ ) ⊗ (cid:101) ξ , α := ρ ⊗ ( (cid:101) ξ + 2 (cid:101) ξ ) + ( ρ − ρ ) ⊗ ( ξ + ξ ), α := ( ρ − ρ ) ⊗ (cid:101) ξ . Table 23.
Base transformation of the character table of (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
4) = S × C .and it is isomorphic to A × C with Coxeter generators s i = (1 2)(3 5) α , s j =(1 2)(3 4) α , s k = (1 5)(2 3) α , for example. In standard notation, this is theexceptional finite Coxeter group H . Remark . These Coxeter generators are not unique, since Out( A × C ) ∼ = C .There are then two sets of Coxeter generators up to conjugation, the other one givenby conjugation by a suitable g ∈ S \ A , for instance, conjugating by g = (2 4): s (cid:48) i = (1 4)(3 5) α , s (cid:48) j = (1 4)(2 3) α , s (cid:48) k = (1 5)(3 4) α . In the first case s j s k =(1 3 4 2 5), conjugated to (1 2 3 4 5), and in the second case s (cid:48) j s (cid:48) k = (1 3 2 4 5),conjugated to (1 3 2 4 5), which represent different conjugacy classes in A .A character table for the alternating group A reads as follows. A e (1 2 3) (1 2)(3 4) (1 2 3 4 5) (1 3 4 5 2) ξ ξ − − ξ − ξ − √
52 1 −√ ξ − −√
52 1+ √ If we call T A the matrix of coefficients of this table then the character table of∆(2 , ,
5) is given by Table 24, where we have used the following representatives ofthe conjugacy classes in terms of the Coxeter generators (‘ ∼ ’ means ‘conjugate’) e ∼ e α ∼ ( s i s j s k ) (1 2 3) ∼ s i s j (1 2 3) α ∼ s i ( s j s k ) (1 2)(3 4) ∼ s i s k (1 2)(3 4) α ∼ s i (1 2 3 4 5) ∼ s j s k (1 2 3 4 5) α ∼ s i s j s k (1 2 3 5 4) ∼ ( s i s j s k ) (1 2 3 5 4) α ∼ s j s i s k QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 37 A × C e s i s j s i s k s j s k ( s i s j s k ) ( s i s j s k ) s i ( s j s k ) s i s i s j s k s j s i s k ρ ⊗ ξ ρ ⊗ ξ ρ ⊗ ξ T A T A ρ ⊗ ξ ρ ⊗ ξ ρ ⊗ ξ ρ ⊗ ξ ρ ⊗ ξ T A − T A ρ ⊗ ξ ρ ⊗ ξ Table 24.
Character table of (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
5) = A × C . Remark . With the non-conjugate set of Coxeter generators s (cid:48) i , s (cid:48) j , s (cid:48) k (see Re-mark 4), we get the analogous set of representatives except swapping the obvi-ous ones, that is, s j s k ∼ ( s (cid:48) i s (cid:48) j s (cid:48) k ) , ( s i s j s k ) ∼ s (cid:48) j s (cid:48) k , ( s i s j s k ) s j s k ∼ s (cid:48) i s (cid:48) j s (cid:48) k and s i s j s k ∼ ( s (cid:48) i s (cid:48) j s (cid:48) k ) s (cid:48) j s (cid:48) k . On the character table this amounts to interchanging ρ i ⊗ ξ with ρ i ⊗ ξ for i = 1 and 2.For the base transformation required, first note that the character table for thealternating group A above can be transformed as follows, A e (1 2 3) (1 2)(3 4) (1 2 3 4 5) (1 3 4 5 2) ξ (cid:101) ξ := ξ − ξ − − − − (cid:101) ξ := ξ − ξ − ξ − (cid:101) ξ := ξ − ξ + ξ √
52 5 −√ (cid:101) ξ := ξ + ξ − ξ − ξ − A e (1 2 3) (1 2)(3 4) (1 2 3 4 5) (1 3 4 5 2) ξ (cid:101)(cid:101) ξ := (cid:101) ξ − (cid:101) ξ − (cid:101) ξ − − (cid:101) ξ − (cid:101) ξ √
52 5 −√ (cid:101) ξ − Appendix B. Induction homomorphisms
In this Appendix, we compute all possible induction homomorphisms R C ( H ) → R C ( G ) appearing in the Bredon chain complex (6). That is, G is a finite Coxetersubgroup of Γ of rank n generated by n ≤ H isa subgroup of G generated by a subset of exactly n − A × C e s i s j s i s k s j s k ( s i s j s k ) ( s i s j s k ) s i ( s j s k ) s i s i s j s k s j s i s k β β √ −√ √ −√ β − − β √
52 5 −√ √
52 5 −√ β − β − − − − − β β β − − √ − √ β − − − − β := ρ ⊗ ξ , β := ρ ⊗ (cid:101)(cid:101) ξ + 2 ρ ⊗ (cid:101) ξ , β := ρ ⊗ (cid:101) ξ , β := ρ ⊗ (cid:101) ξ , β := ρ ⊗ (cid:101) ξ − ρ − ρ ) ⊗ ξ , β := ( ρ − ρ ) ⊗ ξ , β := ( ρ − ρ ) ⊗ (cid:101)(cid:101) ξ , β := ( ρ − ρ ) ⊗ (cid:101) ξ , β := ( ρ − ρ ) ⊗ (cid:101) ξ , β := ( ρ − ρ ) ⊗ ( (cid:101) ξ + 4 ξ ). Table 25.
Base transformation of the character table of (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
5) = A × C .3 subgroups, as this is needed for the simultaneous base transformation argumentin the proof that H Fin (Γ; R C ) is torsion-free (Section 5).We implicitly use the character tables and notation in Appendix A, and Frobe-nius reciprocity [24], throughout this Appendix. We also note that Frobenius reci-procity extends linearly: Lemma 6. If H is a subgroup of a finite group G and φ and η , respectively τ and π , are representations of G , respectively H , then ( φ ↓ + ξ ↓ | τ + π ) = ( φ + ξ | τ ↑ + π ↑ ) . Proof. ( φ ↓ + ξ ↓ | τ + π ) = 1 | H | (cid:88) h ∈ H ( φ ↓ + ξ ↓ )( h ) · ( τ + π )( h )= 1 | H | (cid:88) h ∈ H ( φ ↓ · τ + ξ ↓ · τ + φ ↓ · π + ξ ↓ · π )( h )which by Frobenius reciprocity on irreducible characters equals1 | G | (cid:88) g ∈ G ( φ · τ ↑ + ξ · τ ↑ + φ · π ↑ + ξ · π ↑ )( h ) = ( φ + ξ | τ ↑ + π ↑ ) . (cid:3) B.1.
Rank 1.
The only induction homomorphism in this case is R C ( { e } ) → R C ( (cid:104) s i (cid:105) )which must the regular representation τ (cid:55)→ ρ + ρ shown, in terms of free abeliangroups, in Figure 9.B.2. Rank 2.
In this case G is a dihedral group with the presentation G = (cid:104) s i , s j | s i = s j = ( s i s j ) m (cid:105) , QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 39 R C ( { e } ) → R C ( C ) Z → Z a (cid:55)→ ( a, a ) Figure 9.
Induction homomorphism from H = { e } to G = (cid:104) s i (cid:105) ∼ = C .where m = m ij and we assume i < j . Consider first the case H = (cid:104) s i (cid:105) . Thecharacters of D m (Table 10) restricted to H are (note that s i = s j ( s i s j ) m − ) D m ↓ e s i χ χ − (cid:99) χ − (cid:99) χ φ p H ∼ = C (Table 1) we obtainthe induced representations ρ ↑ = χ + (cid:99) χ + (cid:80) φ p ,ρ ↑ = χ + (cid:99) χ + (cid:80) φ p . The other case is when H = (cid:104) s j (cid:105) . This is analogous, but note that, in order to keepthe notation consistent with Table 2, the characters χ and χ must be interchangedin the even case. Specifically, we have now s j = s j ( s i s j ) and hence D m ↓ e s j χ χ − (cid:99) χ (cid:99) χ − φ p ρ ↑ = χ + (cid:99) χ + (cid:80) φ p ,ρ ↑ = χ + (cid:99) χ + (cid:80) φ p . All in all, as maps of free abelian groups, we have that the induction homomor-phisms R C ( H ) → R C ( G ) shown in Figure 10. R C ( H ) → R C ( G ) Z → Z c ( D m ) ( a, b ) (cid:55)→ ( a, b, (cid:98) b, (cid:98) a, a + b, . . . , a + b ) for H = (cid:104) s i (cid:105) , ( a, b ) (cid:55)→ ( a, b, (cid:98) a, (cid:98) b, a + b, . . . , a + b ) for H = (cid:104) s j (cid:105) , Figure 10.
Induction homomorphisms from H to G = (cid:104) s i , s j (cid:105) ∼ = D m , m = m ij and i < j .B.3. Rank 3.
We compute each case individually.
B.3.1. G = ∆(2 , , . This group is isomorphic to C × C × C , and has Coxetergenerators s i , s j , s k where i < j < k . We compute the induction homomorphismsfor the Coxeter subgroups (cid:104) s i , s j (cid:105) , (cid:104) s i , s k (cid:105) and (cid:104) s j , s k (cid:105) , all three direct factors of G and isomorphic to C × C . Using the bases of R C ( C × C ) and R C (∆(2 , , C (Tables 10 and 14), we immediately obtain the induction homo-morphisms shown, as maps of free abelian groups, shown in Figure 11. Remark . Recall how to the induction homomorphism works from a group A todirect product A × B : if ρ is a representation of A then Ind A × BA ( ρ ) = ρ ⊗ r B , where r B is the regular representation of B . R C ( H ) → R C ( G ) Z → Z ( a, b, c, d ) (cid:55)→ ( a, a, b, b, c, c, d, d ) for H = (cid:104) s i , s j (cid:105) , ( a, b, c, d ) (cid:55)→ ( a, b, a, b, c, d, c, d ) for H = (cid:104) s i , s k (cid:105) , ( a, b, c, d ) (cid:55)→ ( a, b, c, d, a, b, c, d ) for H = (cid:104) s j , s k (cid:105) . Figure 11.
Induction homomorphisms from H to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
2) = C × C × C , and i < j < k .On the other hand, with respect to the transformed bases (Tables 11 and 15 inAppendix A), the induction homomorphisms take the form shown in Tables 26, 27,and 28. Note that, for the transformed bases, we show the restricted characters,and the induced map, in two adjacent tables separated by double vertical bars. (cid:104) s j , s k (cid:105) (cid:44) → C × C × C e s j s k s j s k ( ·| (cid:80) χ i ) ( ·| χ + χ ) ( ·| χ + χ ) ( ·| χ ) ρ ↓ ρ − ρ ↓ − − ρ − ρ ↓ − − ρ − ρ ↓ − − ρ − ρ ↓ ρ − ρ ↓ − − ρ − ρ ↓ ρ − ρ ↓ − − Table 26.
Restricted characters and induced map (cid:104) s j , s k (cid:105) (cid:44) → ∆(2 , ,
2) = C × C × C .B.3.2. G = ∆(2 , , m ) , m > . This group is isomorphic to C × D m with Coxeterpresentation∆(2 , , m ) = (cid:10) s i , s j , s k | s i , s j , s k , ( s i s j ) , ( s i s k ) , ( s j s k ) m (cid:11) , We have three relevant Coxeter subgroups H , which we treat separately. In eachcase, we restrict the characters of G (Table 16) to the subgroup H and then useFrobenius reciprocity to write the induced characters of H into G in terms of thecharacters of G . QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 41 (cid:104) s i , s k (cid:105) (cid:44) → C × C × C e s i s k s i s k ( ·| (cid:80) χ i ) ( ·| χ + χ ) ( ·| χ + χ ) ( ·| χ ) ρ ↓ ρ − ρ ↓ − − ρ − ρ ↓ ρ − ρ ↓ − − ρ − ρ ↓ − − ρ − ρ ↓ − − ρ − ρ ↓ − − ρ − ρ ↓ − − Table 27.
Restricted characters and induced map (cid:104) s i , s k (cid:105) (cid:44) → ∆(2 , ,
2) = C × C × C . (cid:104) s i , s j (cid:105) (cid:44) → C × C × C e s i s j s i s j ( ·| (cid:80) χ i ) ( ·| χ + χ ) ( ·| χ + χ ) ( ·| χ ) ρ ↓ ρ − ρ ) ↓ ρ − ρ ) ↓ − − ρ − ρ ) ↓ ρ − ρ ) ↓ − − ρ − ρ ) ↓ ρ − ρ ) ↓ − ρ − ρ ) ↓ Table 28.
Restricted characters and induced map D ∼ = (cid:104) s i , s j (cid:105) (cid:44) → ∆(2 , ,
2) = C × C × C .Case 1: H = (cid:104) s i , s j (cid:105) ∼ = D = C × C .The elements e, s i , s j and s i s j of H are obtained from the 1st, 3rd, 2nd and 4thcolumn of Table 16 for r equals 0 , , n − n − C × D m ↓ e s i s j s i s j ρ ⊗ χ ρ ⊗ χ − − ρ ⊗ (cid:99) χ − − ρ ⊗ (cid:99) χ ρ ⊗ φ p ρ ⊗ χ − − ρ ⊗ χ − − ρ ⊗ (cid:99) χ − − ρ ⊗ (cid:99) χ − − ρ ⊗ φ p − H with those of G ↓ H above, and divide by | H | = 4) to obtain the coefficientsof the induced irreducible representations of H in G in terms of the irreduciblerepresentations of G . If i < j then the character table of H is the one given inTable 10, but if j < i we should interchange the second and third characters (rows) in that table. This gives ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p if i < j , and ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p if j < i . Equivalently, as a map of free abelian groups, we have the homomorphismsshown in Figure 12. R C ( H ) → R C ( G ) Z → Z · c ( D m ) ( a, b, c, d ) (cid:55)→ ( a, b, (cid:98) b, (cid:98) a, a + b, . . . , a + b, c, d, (cid:98) d, (cid:98) c, c + d, . . . , c + d ) if i < j, ( a, b, c, d ) (cid:55)→ ( a, c, (cid:98) c, (cid:98) a, a + c, . . . , a + c, b, d, (cid:98) d, (cid:98) b, b + d, . . . , b + d ) if j < i. Figure 12.
Induction homomorphisms from H = (cid:104) s i , s j (cid:105) ∼ = C × C to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , , m ) = C × D m , m = m jk >
2, and j < k .Case 2: H = (cid:104) s i , s k (cid:105) ∼ = C × C .Restricting Table 16 to H (1st, 3rd, 2nd and 4th column for k equals 0) we get C × D m ↓ e s i s k s i s k ρ ⊗ χ ρ ⊗ χ − − ρ ⊗ (cid:99) χ ρ ⊗ (cid:99) χ − − ρ ⊗ φ p ρ ⊗ χ − − ρ ⊗ χ − − ρ ⊗ (cid:99) χ − − ρ ⊗ (cid:99) χ − − ρ ⊗ φ p − i < k , ρ ⊗ ρ ⊗ ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p ρ ⊗ ρ ↑ = ρ ⊗ χ + (cid:92) ρ ⊗ χ + (cid:80) ρ ⊗ φ p QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 43 If k < i , the calculation is the same but we must interchange again the 2nd and3rd generators. All in all, we have the homomorphisms shown in Figure 13 as mapsbetween free abelian groups. R C ( H ) → R C ( G ) Z → Z · c ( D m ) ( a, b, c, d ) (cid:55)→ ( a, b, (cid:98) a, (cid:98) b, a + b, . . . , a + b, c, d, (cid:98) c, (cid:98) d, c + d, . . . , c + d ) if i < k, ( a, b, c, d ) (cid:55)→ ( a, c, (cid:98) a, (cid:98) c, a + c, . . . , a + c, b, d, (cid:98) b, (cid:98) d, b + d, . . . , b + d ) if k < i. Figure 13.
Induction homomorphisms from H = (cid:104) s i , s k (cid:105) ∼ = C × C to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , , m ) = C × D m , m = m jk >
2, and j < k .Case 3: H = (cid:104) s j , s k (cid:105) ∼ = D m In this case the condition j < k already holds. Restricting Table 16 to H (the firsttwo columns) C × D m ↓ ( s j s k ) r s k ( s j s k ) r ρ ⊗ χ ρ ⊗ χ ρ ⊗ (cid:99) χ T m ρ ⊗ (cid:99) χ ρ ⊗ φ p ρ ⊗ χ ρ ⊗ χ ρ ⊗ (cid:99) χ T m ρ ⊗ (cid:99) χ ρ ⊗ φ p where T m are the coefficients of the character table of D m as in Table 2. Thisimmediately gives χ i ↑ = ρ ⊗ χ i + ρ ⊗ χ i for all i , and φ p ↑ = ρ ⊗ φ i + ρ ⊗ φ p for all p .Equivalently, as a map of free abelian groups, this induction homomorphism is theone given in Figure 14. R C ( H ) → R C ( G ) Z c ( D m ) → Z · c ( D m ) ( a, b, (cid:98) c, (cid:98) d, r , . . . , r N ) (cid:55)→ ( a, b, (cid:98) c, (cid:98) d, r , . . . , r N , a, b, (cid:98) c, (cid:98) d, r , . . . , r N ) . Figure 14.
Induction homomorphisms from H = (cid:104) s j , s k (cid:105) ∼ = D m to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , , m ) = C × D m , m = m jk >
2, and j < k . Finally, we compute the induction homomorphisms with respect to the trans-formed basis (Tables 11, 12, 13, 17, 18, 19 in Appendix A), summarising the resultsin Tables 29 to 34. D m (cid:44) → D m × C s i ( s i s j ) r α α α β k ρ ⊗ χ ↓ ρ ⊗ ( χ − χ ) ↓ − ρ ⊗ ( φ − χ − χ ) ↓ b r ρ ⊗ ( φ p − φ p − ) ↓ ... 0 a p,r δ p,k ( ρ − ρ ) ⊗ χ ↓ ρ − ρ ) ⊗ ( χ − χ ) ↓ ρ − ρ ) ⊗ ( φ − χ − χ ) ↓ ρ − ρ ) ⊗ ( φ p − φ p − ) ↓ ... 0 0 0 0 0 0where α := (cid:80) (cid:96) =1 χ (cid:96) + 2 ( m − / (cid:80) (cid:96) =1 φ (cid:96) , α := χ + ( m − / (cid:80) (cid:96) =1 φ (cid:96) , α := ( m − / (cid:80) (cid:96) =1 φ (cid:96) , β k := ( m − / (cid:80) (cid:96) = k φ (cid:96) . Table 29.
Restricted characters and map induced by D m (cid:44) → ∆(2 , , m ) = D m × C for m ≥ a p,r := 2 cos( πprm ) − π ( p − rm ), b r := 2 cos( πrm ) − δ p,k the Kronecker delta,2 ≤ p, k ≤ m − and 0 ≤ r ≤ m − . D (cid:44) → D m × C e s i α αs i (cid:80) χ i χ + (cid:99) χ (cid:99) χ (cid:99) χ + (cid:99) χ ρ ⊗ χ ↓ ρ ⊗ ( χ − χ ) ↓ − − ρ ⊗ ( φ − χ − χ ) ↓ ρ ⊗ ( φ p − φ p − ) ↓ ... 0 0 0 0 0 0 0 0( ρ − ρ ) ⊗ χ ↓ − − ρ − ρ ) ⊗ ( χ − χ ) ↓ ρ − ρ ) ⊗ ( φ − χ − χ ) ↓ ρ − ρ ) ⊗ ( φ p − φ p − ) ↓ ... 0 0 0 0 0 0 0 0 Table 30.
Restricted characters and map induced by the two in-clusions D (cid:44) → ∆(2 , , m ) = D m × C , for m ≥ ≤ p ≤ m − .B.3.3. G = ∆(2 , , . This group is isomorphic to the symmetric group S withCoxeter presentation(13) ∆(2 , ,
3) = (cid:10) s i , s j , s k | s i , s j , s k , ( s i s j ) , ( s i s k ) , ( s j s k ) (cid:11) and we assume i < k . We have again three relevant Coxeter subgroups H .Case 1: H = (cid:104) s i , s j (cid:105) ∼ = D D m (cid:44) → D m × C s i ( s i s j ) r s j s i s j α α α α α β k ρ ⊗ χ ↓ ρ ⊗ ( χ − χ ) ↓ − − ρ ⊗ ( χ − χ ) ↓ c r − − ρ ⊗ ( χ − χ ) ↓ − c r ρ ⊗ ( φ − χ − χ ) ↓ − b r ρ ⊗ ( φ p − φ p − ) ↓ ... 0 a p,r δ p,k ( ρ − ρ ) ⊗ χ ↓ ( ρ − ρ ) ⊗ ( χ − χ ) ↓ ( ρ − ρ ) ⊗ ( χ − χ ) ↓ ( ρ − ρ ) ⊗ ( χ + χ − χ − χ ) ↓ ( ρ − ρ ) ⊗ ( φ − χ − χ ) ↓ ... ( ρ − ρ ) ⊗ ( φ p − φ p − ) ↓ ...where α := (cid:80) (cid:96) =1 χ (cid:96) + 2 m − (cid:80) p φ p , α := χ + χ + m − (cid:80) (cid:96) =1 φ (cid:96) , α := χ + m − (cid:80) (cid:96) =1 φ (cid:96) , α := χ + χ + m − (cid:80) (cid:96) =1 φ (cid:96) , α := m − (cid:80) (cid:96) =1 φ (cid:96) , β k := m − (cid:80) (cid:96) = k φ (cid:96) . Table 31.
Restricted characters and map induced by the inclusion D m (cid:44) → ∆(2 , , m ) = D m × C , for m ≥ a p,r := 2 cos( πprm ) − π ( p − rm ), b r := 2 cos( πrm ) − ( − r − c r := ( − r − δ p,k the Kronecker delta, 2 ≤ p, k ≤ m −
1, and0 ≤ r ≤ m . D (cid:44) → D m × C e s j s i s j α αs j s i s j (cid:80) χ i χ + (cid:99) χ (cid:99) χ (cid:99) χ + (cid:99) χ ρ ⊗ χ ↓ ρ ⊗ ( χ − χ ) ↓ − − ρ ⊗ ( χ − χ ) ↓ ρ ⊗ ( χ − χ ) ↓ ρ ⊗ ( φ − χ − χ ) ↓ ρ ⊗ ( φ p − φ p − ) ↓ ... 0 0 0 0 0 0 0 0( ρ − ρ ) ⊗ χ ↓ − − ρ − ρ ) ⊗ ( χ − χ ) ↓ ρ − ρ ) ⊗ ( χ − χ ) ↓ ρ − ρ ) ⊗ ( χ + χ − χ − χ ) ↓ ρ − ρ ) ⊗ ( φ − χ − χ ) ↓ ρ − ρ ) ⊗ ( φ p − φ p − ) ↓ ... 0 0 0 0 0 0 0 0 Table 32.
Restricted characters and map induced by the two in-clusions D (cid:44) → ∆(2 , , m ) = D m × C , for m ≥ ≤ p ≤ m −
1. Our choice on the generatorsis (12)(34) (cid:55)→ s j s i s j and (12) (cid:55)→ α ; any other choice yields anequivalent matrix.The expanded character table for D (from Table 2), assuming first i < j , is D e s i s j s i s j s j s i s i s j s i χ χ − − − φ − − D m (cid:44) → D m × C s i ( s i s j ) r s j s i s j α α α α α β k ρ ⊗ χ ↓ ρ ⊗ ( χ − χ ) ↓ − − ρ ⊗ ( χ − χ ) ↓ c r − ρ ⊗ ( χ − χ ) ↓ c r − ρ ⊗ ( φ − χ − χ ) ↓ b r ρ ⊗ ( φ p − φ p − ) ↓ ... 0 a p,r δ p,k ( ρ − ρ ) ⊗ χ ↓ ( ρ − ρ ) ⊗ ( χ − χ ) ↓ ( ρ − ρ ) ⊗ ( χ − χ ) ↓ ( ρ − ρ ) ⊗ ( χ + χ − χ − χ ) ↓ ( ρ − ρ ) ⊗ ( φ − χ − χ ) ↓ ... ( ρ − ρ ) ⊗ ( φ p − φ p − ) ↓ ...where α := (cid:80) (cid:96) =1 χ (cid:96) + 2 m − (cid:80) p φ p , α := χ + χ + m − (cid:80) (cid:96) =1 φ (cid:96) , α := χ + m − (cid:80) (cid:96) =1 φ (cid:96) , α := χ + χ + m − (cid:80) (cid:96) =1 φ (cid:96) , α := m − (cid:80) (cid:96) =1 φ (cid:96) , β k := m − (cid:80) (cid:96) = k φ (cid:96) . Table 33.
Restricted characters and map induced by the inclusion D m (cid:44) → ∆(2 , , m ) = D m × C , for m ≥ a p,r :=2 cos( πprm ) − π ( p − rm ), b r := 2 cos( πrm ) − c r := ( − r − δ p,k the Kronecker delta, 2 ≤ p, k ≤ m − < r < m and1 < k < m − D (cid:44) → D m × C e s i α αs i (cid:80) χ i χ + (cid:99) χ (cid:99) χ (cid:99) χ + (cid:99) χ ρ ⊗ χ ↓ ρ ⊗ ( χ − χ ) ↓ − − ρ ⊗ ( χ − χ ) ↓ ρ ⊗ ( χ − χ ) ↓ ρ ⊗ ( φ − χ − χ ) ↓ ρ ⊗ ( φ p − φ p − ) ↓ ... 0 0 0 0 0 0 0 0( ρ − ρ ) ⊗ χ ↓ − − ρ − ρ ) ⊗ ( χ − χ ) ↓ ρ − ρ ) ⊗ ( χ − χ ) ↓ ρ − ρ ) ⊗ ( χ + χ − χ − χ ) ↓ ρ − ρ ) ⊗ ( φ − χ − χ ) ↓ ρ − ρ ) ⊗ ( φ p − φ p − ) ↓ ... 0 0 0 0 0 0 0 0 Table 34.
Restricted characters and map induced by the twoinclusions D (cid:44) → ∆(2 , , m ) = D m × C , for m ≥ ≤ p ≤ m −
1. Our choice on the generators is (12)(34) (cid:55)→ s i and (12) (cid:55)→ α , and any other choice yields an equivalent matrix.There are 3 conjugacy classes, { e } , { s i , s j , s i s j s i = s j s i s j } and { s i s j , s j s i } , whichremain unchanged if we swap s i and s j , hence if j < i the table stays the sameand we do not have to treat those two cases separately. The character table of G QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 47 (Table 20) restricted to H consists on the 1st, 2nd, 2nd, 3rd, 3rd, 2nd columns(since s i ∼ s j , s j s i ∼ s i s j and s i s j s i ∼ s i ): S ↓ e s i s j s i s j s j s i s i s j s i ξ ξ − − − ξ − − ξ ξ − − − | H | = 6 we obtain χ ↑ = ξ + ξ χ ↑ = ξ + ξ φ ↑ = ξ + ξ + ξ Equivalently, we have the map of free abelian groups shown in Figure 15. R C ( H ) → R C ( G ) Z → Z ( a, b, c ) (cid:55)→ ( a, b, c, a + c, b + c ) . Figure 15.
Induction homomorphism from H = (cid:104) s i , s j (cid:105) ∼ = D or H = (cid:104) s j , s k (cid:105) ∼ = D to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
3) = S , and i < k .Case 2: H = (cid:104) s j , s k (cid:105) ∼ = D This case is completely analogous to the previous one, so we obtain the same map,also shown in Figure 15.Case 3: H = (cid:104) s i , s k (cid:105) ∼ = D = C × C The character table of G (Table 20) restricted to H consists on the 1st, 2nd, 2nd,5th columns: S ↓ e s i s k s i s k ξ ξ − − ξ ξ − ξ − − − H (Table 10) and dividing by | H | = 4we obtain (note that i < k already holds) ρ ⊗ ρ ↑ = ξ + ξ + ξ ρ ⊗ ρ ↑ = ξ + ξ ρ ⊗ ρ ↑ = ξ + ξ ρ ⊗ ρ ↑ = ξ + ξ + ξ Equivalently, this is the homomorphism of abelian groups shown in Figure 16. Notethat this is the first time that of an induction homomorphism with nontrivial kernel.Now we give the induction homomorphisms with respect to the transformedbases (Tables 11, 12, 21 in Appendix A), summarised in Table 35. R C ( H ) → R C ( G ) Z → Z ( a, b, c, d ) (cid:55)→ ( a, d, a + d, a + b + c, b + c + d ) . Figure 16.
Induction homomorphism from H = (cid:104) s i , s k (cid:105) ∼ = C × C to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
3) = S , and i < k . D (cid:44) → S (1) (1 2) (1 2)(3 4) ( (cid:101) ξ i | (cid:80) χ i ) ( (cid:101) ξ i | χ + (cid:99) χ ) ( (cid:101) ξ i | (cid:99) χ ) ( (cid:101) ξ i | (cid:99) χ ) (cid:101) ξ ↓ (cid:101) ξ ↓ − (cid:101) ξ ↓ (cid:101) ξ ↓ − (cid:101) ξ ↓ D (cid:44) → S (1) (12) (123) ( (cid:101) ξ i | φ + (cid:80) χ i ) ( (cid:101) ξ i | χ + φ ) ( (cid:101) ξ i | φ ) (cid:101) ξ ↓ (cid:101) ξ ↓ − (cid:101) ξ ↓ − (cid:101) ξ ↓ (cid:101) ξ ↓ (cid:101) ξ := ξ , (cid:101) ξ := ξ − ξ , (cid:101) ξ := ξ − ξ − ξ , (cid:101) ξ := ξ − ξ − ξ , (cid:101) ξ := ξ − ξ − ξ + ξ . Table 35.
Restricted characters and map induced by the inclu-sions of D and D into S . The second inclusion (cid:104) (23) , (34) (cid:105) ∼ = D (cid:44) → S induces the same map as the first one, because (23) ∼ (12) and (234) ∼ (123).B.3.4. G = ∆(2 , , . This group is isomorphic to S × C with Coxeter presenta-tion ∆(2 , ,
4) = (cid:10) s i , s j , s k | s i , s j , s k , ( s i s j ) , ( s i s k ) , ( s j s k ) (cid:11) . The three relevant induction homomorphism are as follows.Case 1: H = (cid:104) s i , s k (cid:105) ∼ = D = C × C The character table of G (Table 22) restricted to H consists on the 1st, 7th, 10th QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 49 and 2nd columns: S × C ↓ e s i s k s i s k ρ ⊗ ξ ρ ⊗ ξ − − ρ ⊗ ξ ρ ⊗ ξ − ρ ⊗ ξ − − − ρ ⊗ ξ − − ρ ⊗ ξ − − ρ ⊗ ξ − ρ ⊗ ξ − ρ ⊗ ξ − i < k . Multiplying these rows with the rows of Table 10 wededuce that (note the shortcut in notation) ρ ⊗ ρ ⊗ ρ ⊗ ρ ↑ = ξ + ξ + ξ + ξ + ξ ρ ⊗ ρ ↑ = ξ + ξ + ξ + ξ + ξ ρ ⊗ ρ ↑ = ξ + ξ + ξ + ξ + ξ ρ ⊗ ρ ↑ = ξ + ξ + ξ + ξ + ξ On the other hand, if k < i , then we should interchange the 2nd and 3rd generators.All in all, we have the homomorphisms of free abelian groups shown in Figure 17. R C ( H ) → R C ( G ) Z → Z ( a, b, c, d ) (cid:55)→ ( a, c, a + c, a + b + d, b + c + d, d, b, b + d, a + c + d, a + b + c ) if i < k, ( a, b, c, d ) (cid:55)→ ( a, b, a + b, a + c + d, b + c + d, d, c, c + d, a + b + d, a + b + c ) if k < i. Figure 17.
Induction homomorphism from H = (cid:104) s i , s k (cid:105) ∼ = C × C to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
4) = S × C .Case 2: H = (cid:104) s i , s j (cid:105) ∼ = D The characters of G (Table 22) restricted to H consists on the 1st, 7th, 7th, 3rd,3rd, 7th columns: S × C ↓ e s i s j s i s j s j s i s i s j s i ρ ⊗ ξ ρ ⊗ ξ − − − ρ ⊗ ξ − − ρ ⊗ ξ ρ ⊗ ξ − − − ρ ⊗ ξ − − − ρ ⊗ ξ ρ ⊗ ξ − − ρ ⊗ ξ − − − ρ ⊗ ξ Multiplying these rows with the rows of character table of D above we deduce that(recall that this table is invariant under interchanging the Coxeter generators) ρ ⊗ ρ ⊗ χ ↑ = ξ + ξ + ξ + ξ χ ↑ = ξ + ξ + ξ + ξ φ ↑ = ξ + ξ + ξ + ξ + ξ + ξ or, equivalently, the linear map shown in Figure 18. R C ( H ) → R C ( G ) Z → Z ( a, b, c ) (cid:55)→ ( a, b, c, a + c, b + c, b, a, c, b + c, a + c ) . Figure 18.
Induction homomorphism from H = (cid:104) s i , s j (cid:105) ∼ = D to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
4) = S × C .Case 3: H = (cid:104) s j , s k (cid:105) ∼ = D First we expand the character table of D , assuming j < k , D e s j s k s j s k s k s j s j s k s j s k s j s k s j s k s j s k χ χ − − − − χ − − − − χ − − − − φ − k < j we should interchange the characters χ and χ in order tomaintain the notation consistent. The characters of G restricted to H are the 1st,7th, 10th, 4th, 4th, 10th, 7th, 5th columns of Table 22: S × C ↓ e s s s s s s s s s s s s s s s s ρ ⊗ ξ ρ ⊗ ξ − − − − ρ ⊗ ξ ρ ⊗ ξ − − − − − ρ ⊗ ξ − − − − − ρ ⊗ ξ − − − − ρ ⊗ ξ − − − − ρ ⊗ ξ − − ρ ⊗ ξ − − − − − ρ ⊗ ξ − QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 51
Multiplying these rows with the rows of the character table of D above we deducethat ρ ⊗ ρ ⊗ χ ↑ = ξ + ξ + ξ χ ↑ = ξ + ξ + ξ χ ↑ = ξ + ξ + ξ χ ↑ = ξ + ξ + ξ φ ↑ = ξ + ξ + ξ + ξ The computation in the case k < j is identical, but interchanging χ and χ . All inall, we have the induction homomorphisms shown, as maps between abelian groups,in Figure X R C ( H ) → R C ( G ) Z → Z ( a, b, c, d, e ) (cid:55)→ ( a, c, a + c, d + e, b + e, b, d, b + d, c + e, a + e ) if j < k, ( a, b, c, d, e ) (cid:55)→ ( a, d, a + d, c + e, b + e, b, c, b + c, d + e, a + e ) if k < j. Figure 19.
Induction homomorphism from H = (cid:104) s j , s k (cid:105) ∼ = D to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
4) = S × C .On the other hand, the induction homomorphisms with respect to the trans-formed bases (Tables 12, 13, 23 in Appendix A) are summarised in Tables 36, 37and 38. D (cid:44) → S × C (1) α (12) α (12)(34) (34) ( ·| (cid:80) χ i ) ( ·| χ + (cid:99) χ ) ( ·| (cid:99) χ + (cid:99) χ ) ( ·| (cid:99) χ ) α ↓ α ↓ − − α ↓ α ↓ − − α ↓ α ↓ − α ↓ − α ↓ α ↓ α ↓ Table 36.
Restricted characters and map induced by the inclusionof D into S × C . Here α , . . . , α are as in Table 23.B.3.5. G = ∆(2 , , . This group is isomorphic to A × C with Coxeter presenta-tion ∆(2 , ,
5) = (cid:10) s i , s j , s k | s i , s j , s k , ( s i s j ) , ( s i s k ) , ( s j s k ) (cid:11) . We have again three relevant induction homomorphisms.Case 1: H = (cid:104) s i , s k (cid:105) ∼ = D = C × C
22 LAFONT, ORTIZ, RAHM, AND S´ANCHEZ-GARC´IA D (cid:44) → S × C (1) α (13) (13)(24) α (12)(34) (1432) α ↓ α ↓ − − α ↓ α ↓ − − α ↓ α ↓ − α ↓ α ↓ α ↓ α ↓ D (cid:44) → S × C ( ·| φ + (cid:80) χ i ) ( ·| χ + (cid:99) χ + φ ) ( ·| (cid:99) χ + φ ) ( ·| (cid:99) χ + χ ) ( ·| φ ) α ↓ α ↓ α ↓ α ↓ − α ↓ − α ↓ α ↓ α ↓ α ↓ α ↓ Table 37.
Restricted characters (top) and map induced by theinclusion of D into S × C (bottom). Here α , . . . , α are as inTable 23. D (cid:44) → S × C (1) α (12) (123) ( ·| φ + (cid:80) χ i ) ( ·| χ + φ ) ( ·| φ ) α ↓ α ↓ − α ↓ − α ↓ α ↓ α ↓ α ↓ α ↓ α ↓ α ↓ Table 38.
Restricted characters and map induced by the inclusionof D into S × C . Here α , . . . , α are as in Table 23.The characters of G (Table 24) restricted to H consists on the 1st, 8th, 8th, 3rd QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 53 columns A × C ↓ e s i s k s i s k ρ ⊗ ξ ρ ⊗ ξ ρ ⊗ ξ ρ ⊗ ξ − − − ρ ⊗ ξ − − − ρ ⊗ ξ − − ρ ⊗ ξ ρ ⊗ ξ − − ρ ⊗ ξ − ρ ⊗ ξ − i < k . Multiplying these rows with the rows of Table 10 we obtain ρ ⊗ ρ ⊗ ρ ⊗ ρ ↑ = ξ + ξ + 2 ξ + ξ + ξ + ξ + ξ ρ ⊗ ρ ↑ = ξ + ξ + ξ + ξ + ξ + ξ + ξ + ξ ρ ⊗ ρ ↑ = ξ + ξ + ξ + ξ + ξ + ξ + ξ + ξ ρ ⊗ ρ ↑ = ξ + ξ + ξ + ξ + ξ + ξ + 2 ξ If k < i we must interchange the 2nd and 3rd generators, but note that we obtainthe same map. All in one, we have one induction map, given as a homomorphismof free abelian groups in Figure 20. R C ( H ) → R C ( G ) Z → Z ( a, b, c, d ) (cid:55)→ ( a, b + c + d, a + b + c + d, b + c + d, b + c + d,d, a + b + c, a + b + c + 2 d, a + b + c, a + b + c ) . Figure 20.
Induction homomorphism from H = (cid:104) s i , s k (cid:105) ∼ = C × C to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
5) = A × C .Case 2: H = (cid:104) s i , s j (cid:105) ∼ = D The characters of G (Table 24) restricted to H consists on the 1st, 8th, 8th, 2nd,2nd and 8th columns: A × C ↓ e s i s j s i s j s j s i s i s j s i ρ ⊗ ξ ρ ⊗ ξ ρ ⊗ ξ − − ρ ⊗ ξ − − − ρ ⊗ ξ − − − ρ ⊗ ξ − − − ρ ⊗ ξ ρ ⊗ ξ − − − − − ρ ⊗ ξ ρ ⊗ ξ Multiplying these rows with the rows of the character table of D (which is inde-pendent of whether i < j or j < i ) ρ ⊗ ρ ⊗ χ ↑ = ξ + ξ + ξ + ξ + ξ + ξ χ ↑ = ξ + ξ + ξ + ξ + ξ + ξ φ ↑ = ξ + 2 ξ + ξ + ξ + ξ + 2 ξ + ξ + ξ This is then the map of free abelian groups shown in Figure 21. R C ( H ) → R C ( G ) Z → Z ( a, b, c ) (cid:55)→ ( a, a + b + c, a + 2 c, b + c, b + c,b, a + b + c, b + 2 c, a + c, a + c ) . Figure 21.
Induction homomorphism from H = (cid:104) s i , s j (cid:105) ∼ = D to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
5) = A × C . D (cid:44) → A × C (1) (12)(35) α (15)(23) α (13)(25) β ↓ β ↓ β ↓ β ↓ β ↓ − β ↓ − − β ↓ β ↓ β ↓ β ↓ D (cid:44) → A × C ( ·| (cid:80) χ i ) ( ·| χ + χ ) ( ·| χ + χ ) ( ·| χ ) β ↓ β ↓ β ↓ β ↓ β ↓ β ↓ β ↓ β ↓ β ↓ β ↓ Table 39.
Restricted characters (top) and map induced by theinclusion of D into A × C (bottom).Case 3: H = (cid:104) s j , s k (cid:105) ∼ = D First we expand the character table for D (from Table 2) D e s j s k s j s k s k s j s j s k s j s k s j s k s j s k s j s k s k s j s k s j s j s k s j s k s j χ χ − − − − − φ ϕ − ϕ − ϕ ϕ φ ϕ ϕ ϕ − ϕ − QUIVARIANT K-HOMOLOGY FOR HYPERBOLIC REFLECTION GROUPS 55 where ϕ is the golden ratio √ , and we have used 2 cos (cid:0) π (cid:1) = ϕ − (cid:0) π (cid:1) = ϕ . Note that this table is independent of interchanging s i with s j .Next we restrict the characters of G (Table 24) to H , that is, the 1st, 8th, 8th, 4th,4th, 8th, 8th, 5th, 5th and 8th columns. A × C ↓ e s j s k s j s k s k s j s j s k s j s k s j s k s j s k s j s k s k s j s k s j s j s k s j s k s j ρ ⊗ ξ ρ ⊗ ξ − − − − ρ ⊗ ξ ρ ⊗ ξ − − ϕ ϕ − − − ϕ + 1 − ϕ + 1 − ρ ⊗ ξ − − − ϕ + 1 − ϕ + 1 − − ϕ ϕ − ρ ⊗ ξ − − − − − ρ ⊗ ξ − − − − ρ ⊗ ξ − − − − − ρ ⊗ ξ ϕ ϕ − ϕ + 1 − ϕ + 1 1 ρ ⊗ ξ − ϕ + 1 − ϕ + 1 1 1 ϕ ϕ Remark . With a non-conjugated choice of Coxeter generators, it would be the5th instead of the 4th column, or, equivalently, swapping ρ i ⊗ ξ with ρ i ⊗ ξ for i = 1 and 2.Multiplying these rows with the rows of character table of D above, and usingthat ϕ − ϕ = 1, we obtain ρ ⊗ ρ ⊗ χ ↑ = ξ + ξ + ξ + ξ χ ↑ = ξ + ξ + ξ + ξ φ ↑ = ξ + ξ + ξ + ξ + ξ + ξ φ ↑ = ξ + ξ + ξ + ξ + ξ + ξ This gives the homomorphism of free abelian groups in Figure 22. R C ( H ) → R C ( G ) Z → Z ( a, b, c, d ) (cid:55)→ ( a, c + d, a + c + d, b + c, b + d,b, c + d, b + c + d, a + c, a + d ) . Figure 22.
Induction homomorphism from H = (cid:104) s j , s k (cid:105) ∼ = D to G = (cid:104) s i , s j , s k (cid:105) ∼ = ∆(2 , ,
5) = A × C .We finish by giving the same induction homomorphisms but this time with re-spect to the transformed bases (Tables 11, 12, 25 in Appendix A) in Tables 39, 40and 41. D (cid:44) → A × C (1) (12)(34) α (123) ( ·| (cid:80) χ i + 2 φ ) ( ·| χ + φ ) ( ·| φ ) β ↓ β ↓ β ↓ − β ↓ β ↓ β ↓ − β ↓ β ↓ β ↓ β ↓ Table 40.
Restricted characters and map induced by the inclusionof D into A × C . D (cid:44) → A × C (1) (12345) (12354) (12)(34) α ( ·| (cid:80) χ i + 2 (cid:80) φ i ) ( ·| χ + (cid:80) φ i ) ( ·| (cid:80) φ i ) ( ·| φ ) β ↓ β ↓ √ −√ β ↓ β ↓ √
52 5 −√ − β ↓ β ↓ − β ↓ β ↓ β ↓ β ↓ Table 41.
Restricted characters and map induced by the inclusionof D into A × C2