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Wolfgang Lueck

Publications and source records attributed to Wolfgang Lueck.

At least 73 records · Page 4Linked to original sources
arXiv2009

On the classifying space of the family of finite and of virtually cyclic subgroups for CAT(0)-groups

Let G be a discrete group which acts properly and isometrically on a complete CAT(0)-space X. Consider an integer d with d=1 or d greater or equal to 3 such that the topological dimension of X is bounded by d. We show the existence of a G-CW-model E_fin(G) for the classifying space for proper G-actions with dim(E_fin(G)) less or equal to d. Provided that the action is also cocompact, we prove the existence of a G-CW-model E_vcyc(G) for the classifying space of the family of virtually cyclic subgroups such that dim(E_vcyc(G)) is less or equal to d+1.

math.AT↗
arXiv2009

On the classifying space of the family of virtually cyclic subgroups

We study the minimal dimension of the classifying space of the family of virtually cyclic subgroups of a discrete group. We give a complete answer for instance if the group is virtually poly-Z, word-hyperbolic or countable locally virtually cyclic. We give examples of groups for which the difference of the minimal dimensions of the classifying spaces of virtually cyclic and of finite subgroups is -1, 0 and 1, and show in many cases that no other values can occur.

math.AT↗
arXiv2008

L^2-torsion of hyperbolic manifolds of finite volume

Suppose $\bar{M}$ is a compact connected odd-dimensional manifold with boundary, whose interior $M$ comes with a complete hyperbolic metric of finite volume. We will show that the $L^2$-topological torsion of $\bar{M}$ and the $L^2$-analytic torsion of the Riemannian manifold $M$ are equal. In particular, the $L^2$-topological torsion of $\bar{M}$ is proportional to the hyperbolic volume of $M$, with a constant of proportionality which depends only on the dimension and which is known to be nonzero in dimension 3, 5 and 7. In dimension 3 this proves the conjecture Of Lott and Lueck which gives a complete calculation of the $L^2$-topological torsion of compact $L^2$-acyclic 3-manifolds which admit a geometric torus-decomposition. In an appendix we give a counterexample to an extension of the Cheeger-Mueller theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, even if the Euler characteristic of the boundary vanishes. Keywords: L^2-torsion, hyperbolic manifolds, 3-manifolds

dg-ga↗
arXiv2008

Survey on geometric group theory

This article is a survey article on geometric group theory from the point of view of a non-expert who likes geometric group theory and uses it in his own research. The sections are: classical examples, basics about quasiisometry,properties and invariants of groups invariant under quasiisometry, rigidity, hyperbolic spaces and CAT(k)-spaces, the boundary of a hyperbolic space, hyperbolic groups, CAT(0)-groups, classifying spaces for proper actions, measurable group theory, some open problems.

math.GR↗
arXiv2008

Equivariant covers for hyperbolic groups

We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture in algebraic K-theory for every word-hyperbolic group G and every coefficient ring R.

math.GT↗
arXiv2007

On crossed product rings with twisted involutions, their module categories and L-theory

We study the Farrell-Jones Conjecture with coefficients in an additive G-category with involution. This is a variant of the L-theoretic Farrell-Jones Conjecture which originally deals with group rings with the standard involution. We show that this formulation of the conjecture can be applied to crossed product rings R*G equipped with twisted involutions and automatically implies the a priori more general fibered version.

math.KT↗
arXiv2007

On the Farrell-Jones and related Conjectures

These extended notes are based on a series of six lectures presented at the summer school ``Cohomology of groups and algebraic $K$-theory'' which took place in Hangzhou, China from July 1 until July 12 in 2007. They give an introduction to the Farrell-Jones and the Baum-Connes Conjecture.

math.KT↗
arXiv2007

Inheritance of Isomorphism Conjectures under colimits

We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell-Jones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have disproved the Baum-Connes Conjecture with coefficients.

math.KT↗
arXiv2007

On the Farrell-Jones Conjecture and its applications

We present the status of the Farrell-Jones Conjecture for algebraic K-theory for a group G and arbitrary coefficient rings R. We add new groups for which the conjecture is known to be true and study inheritance properties. We discuss new applications, focussing on the Bass Conjecture, the Kaplansky Conjecture and conjectures generalizing Moody's Induction Theorem. Thus we extend the class of groups for which these conjectures are known considerably.

math.KT↗
arXiv2007

Various L2-signatures and a topological L2-signature theorem

For a normal covering over a closed oriented topological manifold we give a proof of the L2-signature theorem with twisted coefficients, using Lipschitz structures and the Lipschitz signature operator introduced by Teleman. We also prove that the L-theory isomorphism conjecture as well as the C^*_max-version of the Baum-Connes conjecture imply the L2-signature theorem for a normal covering over a Poincar space, provided that the group of deck transformations is torsion-free. We discuss the various possible definitions of L2-signatures (using the signature operator, using the cap product of differential forms, using a cap product in cellular L2-cohomology,...) in this situation, and prove that they all coincide.

math.GT↗
arXiv2006

The structure of crossed products of irrational rotation algebras by finite subgroups of SL_2 (Z)

Let F be a finite subgroup of SL_2 (Z) (necessarily isomorphic to one of Z/2Z, Z/3Z, Z/4Z, or Z/6Z), and let F act on the irrational rotational algebra A_θ via the restriction of the canonical action of SL_2 (Z). Then the crossed product of A_θ by F, and the fixed point algebra for the action of F on A_θ, are AF algebras. The same is true for the crossed product and fixed point algebra of the flip action of Z/2Z on any simple d-dimensional noncommutative torus A_Θ. Along the way, we prove a number of general results which should have useful applications in other situations.

math.OA↗
arXiv2005

Induction Theorems and Isomorphism Conjectures for K- and L-Theory

The Farrell-Jones and the Baum-Connes Conjecture say that one can compute the algebraic K- and L-theory of the group ring and the topological K-theory of the reduced group C^*-algebra of a group G in terms of these functors for the virtually cyclic subgroups or the finite subgroups of G. By induction theory we want to reduce these families of subgroups to a smaller family, for instance to the family of subgroups which are either finite hyperelementary or extensions of finite hyperelementary groups with infinite cyclic kernel or to the family of finite cyclic subgroups. Roughly speaking, we extend the induction theorems of Dress for finite groups to infinite groups.

math.KT↗
arXiv2005

Topological rigidity for non-aspherical manifolds

The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f: N --> M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, then there is an orientation preserving homeomorphism h: N --> M such that h and f induce up to conjugation the same maps on the fundamental groups. We call such manifolds Borel manifolds. We give partial answers to this questions for S^k x S^d, for sphere bundles over aspherical closed manifolds of dimension less or equal to 3 and for 3-manifolds with torsionfree fundamental groups. We show that this rigidity is inherited under connected sums in dimensions greater or equal to 5. We also classify manifolds of dimension 5 or 6 whose fundamental group is the one of a surface and whose second homotopy group is trivial.

math.GT↗
arXiv2005

K- and L-theory of the semi-direct product of the discrete 3-dimensional Heisenberg group by Z/4

We compute the group homology, the topological K-theory of the reduced C^*-algebra, the algebraic K-theory and the algebraic L-theory of the group ring of the semi-direct product of the three-dimensional discrete Heisenberg group by Z/4. These computations will follow from the more general treatment of a certain class of groups G which occur as extensions 1-->K-->G-->Q-->1 of a torsionfree group K by a group Q which satisfies certain assumptions. The key ingredients are the Baum-Connes and Farrell-Jones Conjectures and methods from equivariant algebraic topology.

math.KT↗
arXiv2005

Detecting K-theory by cyclic homology

We discuss which part of the rationalized algebraic K-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology.

math.KT↗
arXiv2005

Rational Computations of the Topological K-Theory of Classifying Spaces of Discrete Groups

We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology of G and of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure.

math.KT↗