Equivariant K-homology for hyperbolic reflection groups

Abstract

We compute the equivariant K-homology of the classifying space for proper actions, for cocompact 3-dimensional hyperbolic reflection groups. This coincides with the topological K-theory of the reduced C * -algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group, the associated K-theory groups are torsion-free. As a result we can promote previous rational computations to integral computations. Our proof relies on a new efficient algebraic criterion for checking torsion-freeness of K-theory groups, which could be applied to many other classes of groups.Date: April 4, 2019. 1 arXiv:1707.05133v2 [math.KT] 3 Apr 2018 2 LAFONT, ORTIZ, RAHM, AND SÁNCHEZ-GARCÍAMain Theorem. Let Γ be a cocompact 3-dimensional hyperbolic reflection group, generated by reflections in the side of a hyperbolic polyhedron P ⊂ H 3 . Then, where the integers cf (Γ), χ(C) can be explicitly computed from the combinatorics of the polyhedron P.Here, cf (Γ) denotes the number of conjugacy classes of elements of finite order in Γ, and χ(C) denotes the Euler characteristic of the Bredon chain complex. By a celebrated result of Andre'ev [2], there is a simple algorithm that inputs a Coxeter group Γ, and decides whether or not there exists a hyperbolic polyhedron P Γ ⊂ H 3 which generates Γ. In particular, given an arbitrary Coxeter group, one can easily verify 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 all discrete groups acting on hyperbolic 3-space. For example, 2-torsion occurs in K 0 (C * r (Γ)) whenever Γ is a Bianchi group containing a 2-dihedral subgroup C 2 ×C 2 (see [23]). In fact, the key difficulty in the proof of our Main Theorem lies in showing that these K-theory groups are torsion-free. Some previous integral computations yielded K-theory groups that are torsion-free, though in those papers the torsionfreeness was a consequence of ad-hoc computations. Our second goal is to give a general criterion which explains the lack of torsion, and can be efficiently checked. This allows a systematic, algorithmic approach to the question of whether a Ktheory group is torsion-free.Let us briefly describe the contents of the paper. In Section 2, we provide background material on hyperbolic reflection groups, topological K-theory, and the Baum-Connes Conjecture. We also introduce our main tool, the Atiyah-Hirzebruch type 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 groups H Fin 0 (Γ; R C ) and H Fin 1 (Γ; R C ) respectively. We also explain, using the Γ-action on H 3 , why the homology group H Fin 1 (Γ; R C ) is torsion-free. In contrast, showing that H Fin 0 (Γ; R C ) is torsion-free is much more difficult. In Section 4, we give a geometric proof for this fact in a restricted setting. In Section 5, we give a linear algebraic proof in the general case, inspired by the "representation ring splitting" technique of [23]....

An introduction to torsion subcomplex reduction

“…In this setting, the technique has inspired work beyond the range of arithmetic groups, which has led to formulas for the integral Bredon homology and equivariant K-homology of all compact 3-dimensional hyperbolic reflection groups [23], through a novel criterion for torsion-freeness of equivariant K-homology in a more general framework.…”

On the equivariant K– and KO–homology of some special linear groups

“…In spite of the interest, to date there have been very few computations of K Γ -and KO Γ -homology. Indeed, on the K Γ -homology side of things there are complete calculations for one relator groups [20], NEC groups [18], some Bianchi groups and hyperbolic reflection groups [16,21,22], some Coxeter groups [9,27,28], and SL 3 (Z) [26]. For KO Γ -homology the author is aware of two complete computations; the first, due to Davis and Lück, on a family of Euclidean crystallographic groups [7], and the second, due to Mario Fuentes-Rumí, on simply connected graphs of cyclic groups of odd order and of some Coxeter groups [10].…”

Bredon homology of wallpaper groups

“…Recall that this statement identifies the equivariant K-homology of the space EG with the K-theory of the reduced C * -algebra of G. The conjecture is true for wallpaper groups, as they are solvable (see Section 2.1), and the corresponding values of K * (C * r G) were computed in his thesis by Yang ( [23] (see also , where in particular a little mistake in Yang's results is corrected). Other computations of Bredon homology in the context of Baum-Connes conjecture may be found for example in [20], [11] or [1].…”