arXiv subjects
Publications and source records attributed to Wolfgang Lueck.
Let G be a cocompact lattice in a virtually connected Lie group or the fundamental group of a 3-manifold. We prove the K-theoretic Farrell-Jones Conjecture (up to dimension one) and the L-theoretic Farrell-Jones Conjecture for G, where we allow coefficients in additive G-categories with (involution).
Let X be a finite graph. Let E be the number of its edges and d be its degree. Denote by F_1(X) its first spectral density function which counts the number of eigenvalues less or equal to lambda^2 of the associated Laplace operator. We prove the estimate F_1(X)(lambda) - F_1(X)(0) le 2 cdot E cdot d cdot lambda for 0 le lambda < 1. We explain how this gives evidence for conjectures about approximating Fuglede-Kadison determinants and L^2-torsion.
We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for GL_n(Z).
In this paper we consider the asymptotic behavior of invariants such as Betti numbers, minimal numbers of generators of singular homology, the order of the torsion subgroup of singular homology, and torsion invariants. We will show that all these vanish in the limit if the CW-complex under consideration fibers in a specific way. In particular we will show that all these vanish in the limit if one considers an aspherical closed manifold which admits a non-trivial S^1-action or whose fundamental group contains an infinite normal elementary amenable subgroup. By considering classifying spaces we also get results for groups.
We investigate the algebraic K- and L-theory of the group ring RG, where G is a hyperbolic or virtually finitely generated abelian group and R is an associative ring with unit.
Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas computing the topological K-(co)homology of the classifying space BG up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups G these formulas are sharp. The main new tools we use for the K-theory calculation are a Cocompletion Theorem and Equivariant Universal Coefficient Theorems which are of independent interest. In the case where G is a finite group these theorems reduce to well-known results of Greenlees and Boekstedt.
We present a sufficient condition for groups to satisfy the Farrell-Jones Conjecture in algebraic K-theory and L-theory. The condition is formulated in terms of finite quotients of the group in question and is motivated by work of Farrell-Hsiang.
We compute the topological K-theory of the group C*-algebra C*_r(G) for a group extension Z^n->G->Z/m provided that the conjugation action of Z/m on Z^n is free outside the origin.
Consider the semi-direct product Z^n rtimes Z/m. A conjecture of Adem-Ge-Pan-Petrosyan predicts that the associated Lyndon-Hochschild-Serre spectral sequence collapses. We prove this conjecture provided that the Z/m-action on Z^n is free outside the origin. We disprove the conjecture in general, namely, we give an example with n=6 and m=4, where the second differential does not vanish.
Let Gamma be a semidirect product of the form Z^n rtimes Z/p where p is prime and the Z/p-action on Z^n is free away from the origin. We will compute the topological K-theory of the real and complex group C*-algebra of Gamma and show that Gamma satisfies the unstable Gromov-Lawson-Rosenberg Conjecture. On the way we will analyze the (co-)homology and the topological K-theory of the classifying spaces BGamma and underbar{B}Gamma. The latter is the quotient of the induced Z/p-action on the torus T^n.
Let G be a finite group. We show that the Bass Nil-groups $NK_n(RG)$, $n \in Z$, are generated from the p-subgroups of G by induction maps, certain twisting maps depending on elements in the centralizers of the p-subgroups, and the Verschiebung homomorphisms. As a consequence, the groups $NK_n(RG)$ are generated by induction from elementary subgroups. For $NK_0(ZG)$ we get an improved estimate of the torsion exponent.
We show an invariance result for the L2-torsion of groups under uniform measure equivalence provided a measure-theoretic version of the determinant conjecture holds. The measure-theoretic determinant conjecture is discussed and, for instance, proved for Bernoulli actions of residually amenable groups.
We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.
This article will explore the K- and L-theory of group rings and their applications to algebra, geometry and topology. The Farrell-Jones Conjecture characterizes K- and L-theory groups. It has many implications, including the Borel and Novikov Conjectures for topological rigidity. Its current status, and many of its consequences are surveyed.
We associate to a CAT(0)-space a flow space that can be used as the replacement for the geodesic flow on the sphere tangent bundle of a Riemannian manifold. We use this flow space to prove that CAT(0)-group are transfer reducible over the family of virtually cyclic groups. This result is an important ingredient in our proof of the Farrell-Jones Conjecture for these groups.
We prove an analogue of the Approximation Theorem of L^2-Betti numbers by Betti numbers for arbitrary coefficient fields and virtually torsionfree amenable groups. The limit of Betti numbers is identified as the dimension of some module over the Ore localization of the group ring.
Let G be a torsion-free hyperbolic group and let n > 5 be an integer. We prove that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n-1)-dimensional sphere.
This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical which is a purely homotopy theoretical condition implies many striking results about the geometry and analysis of the manifold or its universal covering, and the ring theoretic properties and the K- and L-theory of the group ring associated to its fundamental group. The Borel Conjecture predicts that closed aspherical manifolds are topologically rigid. The article contains new results about product decompositions of closed aspherical manifolds and an announcement of a result joint with Arthur Bartels and Shmuel Weinberger about hyperbolic groups with spheres of dimension greater or equal to six as boundary. At the end we describe (winking) our universe of closed manifolds.