Abstract.A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in O C .3; 1/, with fundamental domain a geodesic simplex in H 3 (possibly with some ideal vertices). The classification of these groups is known, and there are exactly 9 cocompact examples, and 23 non-cocompact examples. We provide a complete computation of the lower algebraic K-theory of the integral group ring of all the hyperbolic 3-simplex reflection groups.
For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in the faces. Furthermore, this Coxeter group is a lattice inside the isometry group of hyperbolic 3-space, with fundamental domain the original polyhedron P. In this paper, we provide a procedure for computing the lower algebraic K-theory of the integral group ring of such Coxeter lattices in terms of the geometry of the polyhedron P. As an ingredient in the computation, we explicitly calculate some of the lower K-groups of the dihedral groups and the product of dihedral groups with the cyclic group of order two.Comment: 35 pages, 2 figure
Abstract. Every virtually cyclic group G that surjects onto the infinite dihedral group D y contains an index two subgroup P of the form H z a Z. We show that the Waldhausen Nil-group of G vanishes if and only if the Farrell Nil-group of P vanishes.2000 Mathematics Subject Classification: 19D35.
Abstract. We provide splitting formulas for certain Waldhausen Nil-groups. We focus on Waldhausen Nil-groups associated to acylindrical amalgamations Γ = G 1 * H G 2 of groups G 1 , G 2 over a common subgroup H. For these amalgamations, we explain how, provided G 1 , G 2 , Γ satisfy the Farrell-Jones isomorphism conjecture, the Waldhausen Nil-groupscan be expressed as a direct sum of Nil-groups associated to a specific collection of virtually cyclic subgroups of Γ. A special case covered by our theorem is the case of arbitrary amalgamations over a finite group H.
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]....
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