The topological K-theory of certain crystallographic groups

Davis, James F.; Lück, Wolfgang

doi:10.4171/jncg/121

Cited by 22 publications

(44 citation statements)

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“…This forces the number of incident faces to be exactly three. The only finite Coxeter groups acting by reflections on S 2 are the triangle groups ∆(2, 2, m) for some m ≥ 2, ∆(2, 3, 3), ∆ (2,3,4) and ∆ (2,3,5), where we use the notation (2) ∆(p, q, r) = s 1 , s 2 , s 3 | s 2 1 , s 2 2 , s 2 3 , (s 1 s 2 ) p , (s 1 s 3 ) q , (s 2 s 3 ) r . From our compact polyhedron P, we obtain an induced Γ-CW-structure on X = H 3 with:…”

Section: 3mentioning

confidence: 99%

“…, α 10 are as in Table 23. B.3.5. G = ∆ (2,3,5). This group is isomorphic to A 5 × C 2 with Coxeter presentation ∆(2, 3, 5) = s i , s j , s k | s 2 i , s 2 j , s 2 k , (s i s j ) 3 , (s i s k ) 2 , (s j s k ) 5 .…”

Section: Appendix B Induction Homomorphismsmentioning

confidence: 99%

“…For 2-dimensional crystallographic groups, such calculations have been done in M. Yang's thesis [26]. This was subsequently extended to the class of cocompact planar groups by Lück and Stamm [19], and to certain higher dimensional dimensional crystallographic groups by Davis and Lück [5] (see also Langer and Lück [13]). For 3-dimensional groups, Lück [15] completed this calculation for the semi-direct product Hei 3 (Z) Z 4 of the 3-dimensional integral Heisenberg group with a specific action of the cyclic group Z 4 .…”

Section: Introductionmentioning

confidence: 99%

“…Hence by Theorem 5, H Fin 0 (Γ; R C ) is torsion-free for Γ the Heisenberg semidirect product group of Lück's article [15]. [5] consider the semidirect product of Z n with the cyclic p-group Z/p, where the action of Z/p on Z n is given by an integral representation, which is assumed to act freely on the complement of zero. The action of this semidirect product group Γ on EΓ ∼ = R n is crystallographic, with Z n acting by lattice translations, and Z/p acting with a single fixed point.…”

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confidence: 99%

“…7. cf (Γ) and χ(C) from the geometry of P Let Γ be the reflection group of the compact 3-dimensional hyperbolic polyhedron P. In this final section, we compute the number of conjugacy classes of elements of finite order of Γ, cf (Γ), and the Euler characteristic of the Bredon chain complex (5), χ(C), from the geometry of the polyhedron P. This gives us explicit combinatorial formulas for the Bredon homology and equivariant K-theory groups computed in our Main Theorem. 7.1.…”

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confidence: 99%

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Equivariant K-homology for hyperbolic reflection groups

Lafont

Ortiz²,

Rahm³

et al. 2018

The Quarterly Journal of Mathematics

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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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Section: 3mentioning

confidence: 99%

Section: Appendix B Induction Homomorphismsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

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mentioning

confidence: 99%

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Equivariant K-homology for hyperbolic reflection groups

Lafont

Ortiz²,

Rahm³

et al. 2018

The Quarterly Journal of Mathematics

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Manifolds Homotopy Equivalent to Certain Torus Bundles over Lens Spaces

Davis

Lück

2020

Comm Pure Appl Math

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We compute the topological simple structure set of closed manifolds that occur as total spaces of flat bundles over lens spaces S l =.Z=p/ with fiber T n for an odd prime p and l 3 provided that the induced Z=p-action on 1 .T n / D Z n is free outside the origin. To the best of our knowledge this is the first computation of the structure set of a topological manifold whose fundamental group is not obtained from torsionfree and finite groups using amalgamated and HNNextensions. We give a collection of classical surgery invariants such as splitting obstructions and -invariants that decide whether a simple homotopy equivalence from a closed topological manifold to M is homotopic to a homeomorphism. © 2020 Wiley Periodicals LLC Contents 0. Introduction 1. Preliminaries about the Group Z n Ì Z=p 2. Preliminaries about the Farrell-Jones Conjecture 3. Algebraic K-theory of the Group Ring 4. Algebraic L-theory of the Group Ring 5. Structure Sets 6. Periodic Simple Structure Set of BP for a Finite p-group 7. Periodic Simple Structure Set of B 8. Periodic Simple Structure Set of M 9. Geometric Simple Structure Set of M 10. Invariants for Detecting the Structure Set of M 11. Appendix: Open Questions Bibliography

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