arXiv subjects
Publications and source records attributed to Wolfgang Lueck.
We give several examples of finite groups $G$ for which the rank of the tensor product $\mathbb{Z} \otimes_{\mathbb{Z}\mathrm{Aut}(G)}$ Wh$(G)$ is or is not zero. This is motivated by an earlier theorem of the first author, which implies as a special case that when this group has nonzero rank, the Whitehead group of any other group (finite or infinite) that contains $G$ as a normal subgroup is rationally nontrivial.
We construct closed manifolds with vanishing L^2-Betti numbers over every field) which do not virtually fibre over the circle. The class of fundamental groups that occurs is the largest possible, and in many cases the dimension may be taken to be six. We construct aspherical closed manifolds with residually (torsionfree and nilpotent) fundamental groups in all dimensions at least three whose L^2-Betti numbers vanish (over every field) and which do not virtually fibre over the circle. In particular this implies that in Kielak's Theorem about virtually algebraic fibring for RFRS-groups one cannot weaken the condition RFRS to residually (torsionfree and nilpotent.
We systematically study L^2-Betti numbers in zero and prime characteristic and apply them to a conjecture of Wise stating that all towers of a finite 2-complex are non-positive if and only if the second L^2-Betti number vanishes.
We develop the theory of $L^2$-torsion of an automorphism of a group and compute it for every automorphism of a group which is hyperbolic and one-ended relative to a finite collection of virtually polycyclic groups. We also prove a combination formula for the $L^2$-torsion of a group in terms of the $L^2$-torsion of its stabilisers of a sufficiently nice action on a contractible space. We apply it to compute the $L^2$-torsion of a selection of CAT(0) lattices, of many relatively hyperbolic groups and their automorphisms, of higher dimensional graph manifolds, and of handlebody groups.
This is a survey on the Farrell-Jones Conjecture about the algebraic K- and L-theory of groups rings and its applications to algebra, geometry, group theory, and topology.
We extend results for the K-theory of Hecke algebras of reductive $p$-adic groups to completed Kac-Moody groups.
We investigate the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for the algebraic $K$-theory of twisted group rings of a group G with coefficients in a regular ring R or, more generally, with coefficients in a regular additive category. They are known to be isomorphisms rationally. We show that it suffices to invert only those primes p for which G contains a non-trivial finite p-group and p is not invertible in R. The key ingredient is the detection of Nil-terms of a twisted group ring of a finite group F after localizing at p in terms of the p-subgroups of F using Verschiebungs and Frobenius operators. We construct and exploit the structure of a module over the ring of big Witt vectors on the Nil-terms. We analyze the algebraic K-theory of the Hecke algebras of subgroups of reductive p-adic groups in prime characteristic.
We construct certain maps from buildings associated to td-groups to a space closely related to the classifying numerable $G$-space for the family $\mathcal{C}$vcy of covirtually cyclic subgroups. These maps are used in forthcoming paper to study the K-theory of Hecke algebras in the spirit of the Farrell-Jones conjecture.
We compute the algebraic K-theory of the Hecke algebra of a reductive p-adic group G using the fact that the Farrell-Jones Conjecture is known in this context. The main tool will be the properties of the associated Bruhat-Tits building and an equivariant Atiyah-Hirzebruch spectral sequence. In particular the projective class group can be written as the colimit of the projective class groups of the compact open subgroups of G.
We consider the problem of whether, for a given virtually torsionfree discrete group $Γ$, there exists a cocompact proper topological $Γ$-manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. This problem is related to Nielsen Realization. We will make the assumption that the expected manifold model has a zero-dimensional singular set. Then we solve the problem in the case, for instance, that $Γ$ contains a normal torsionfree subgroup $π$ such that $π$ is hyperbolic and $π$ is the fundamental group of an aspherical closed manifold of dimension greater or equal to five and $Γ/π$ is a finite cyclic group of odd order.
Motivated by the Farrell-Jones Conjecture for group rings, we formulate the $\mathcal{C}$op-Farrell-Jones Conjecture for the K-theory of Hecke algebras of td-groups. We prove this conjecture for (closed subgroups of) reductive p-adic groups G. In particular, the projective class group $K_0(\mathcal{H}(G))$ for a (closed subgroup) of a reductive p-adic group G can be computed as a colimit of projective class groups $K_0(\mathcal{H}(U))$ where U varies over the compact open subgroups of G. This implies that all finitely generated smooth complex representations of a reductive p-adic G admit finite projective resolutions by compactly induced representations. For SL$_n(F)$ we translate the colimit formula for $K_0(\mathcal{H}(G))$ to a more concrete cokernel description in terms of stabilizers for the action on the Bruhat-Tits building. For negative K-theory we obtain vanishing results, while we identify the higher K-groups $K_n(\mathcal{H}(G))$ with the value of G-homology theory on the extended Bruhat-Tits building. Our considerations apply to general Hecke algebras of the form $\mathcal{H}(G;R,ρ,ω)$, where we allow a central character $ω$ and a twist by an action $ρ$ of G on R. For the $\mathcal{C}$op-Farrell-Jones Conjecture we need to assume $\mathbb{Q} \subseteq R$ and a regularity assumption. As a key intermediate step we introduce the $\mathcal{C}vcy-Farrell-Jones conjecture. For the latter no regularity assumptions on R are needed.
In this article we give a survey on open problems and conjectures concerning L^2-invariants. We cover the whole portfolio and not only certain aspects as they are considered in the previous more specialized (and within their scope more detailed) survey articles [87,89]. Moreover, we include some new results and problems, which have occurred after these two survey articles were written. The reader may select a specific topic by looking at the table of contents below.
In this paper we formulate and lay the foundations for the K-theoretic Farrell-Jones Conjecture for the Hecke algebra of totally disconnected groups. The main result of his paper is the proof that it passes to closed subgroups. Moreover, we carry out some constructions such as the diagonal tensor product and prove some results that will be used in the actual proof of the Farrell-Jones Conjecture for reductive p-adic groups, which will appear in a different paper.
We extend the notion of regular coherence from rings to additive categories and show that well-known consequences of regular coherence for rings also apply to additive categories. For instance the negative K-groups and all twisted Nil-groups vanish for an additive category if it is regular coherent. This will be applied to nested sequences of additive categories, motivated by our ongoing project to determine the algebraic K-theory of the Hecke algebra of a reductive p-adic group.
Consider a totally disconnected group G, which is covirtually cyclic, i.e., contains a normal compact open subgroup L such that G/L is infinite cyclic. We establish a Wang sequence, which computes the algebraic K-groups of the Hecke algebra of G in terms of the one of L, and show that all negative K-groups vanish. This confirms the K-theoretic Farrell-Jones Conjecture for the Hecke algebra of G in this special case. Our ultimate long term goal is to prove it for any closed subgroup of any reductive p-adic group. The results of this paper will play a role in the final proof.
There is the problem, whether for a given virtually torsionfree discrete group $Γ$ there exists a cocompact proper topological $Γ$-manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. It is related to Nielsen's Realization and to the problem of Brown, whether there is a d-dimensional model for the classifying space for proper actions, if the underlying group has virtually cohomological dimension d. Assuming that the expected manifold model has a zero-dimensional singular set, we solve the problem in the Poincaré category and obtain new results about Brown's problem under certain conditions concerning the underlying group, for instance if it is hyperbolic. In a sequel paper together with James Davis we will deal with this on the level of topological manifolds.
In this paper give a survey about L^2-invariants focusing on 3-manifolds.
We consider the problem whether for a group G there exists a constant Lambda(G) > 1 such that for any (r,s)-matrix A over the integral group ring ZG the Fuglede-Kadison determinant of the G-equivariant bounded operator from L^2(G)^r to L^2(G)^s given by right multiplication with A is either one or greater or equal to Lambda(G). If G is the infinite cyclic group and we consider only r = s = 1, this is precisely Lehmer's problem.