Let G be a torsion free discrete group and let Q denote the field of algebraic numbers in C. We prove that QG fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups which are residually torsion free elementary amenable or which are residually free. This result implies that there are no non-trivial zero-divisors in CG. The statement relies on new approximation results for L 2-Betti numbers over QG, which are the core of the work done in this paper. Another set of results in the paper is concerned with certain number theoretic properties of eigenvalues for the combinatorial Laplacian on L 2-cochains on any normal covering space of a finite CW complex. We establish the absence of eigenvalues that are transcendental numbers, whenever the covering transformation group is either amenable or in the Linnell class C. We also establish the absence of eigenvalues that are Liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class G. MSC: 55N25 (homology with local coefficients), 16S34 (group rings, Laurent rings), 46L50 (non-commutative measure theory)

For a normal covering over a closed oriented topological manifold we give a proof of the L 2-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 L 2-signature theorem for a normal covering over a Poincaré space, provided that the group of deck transformations is torsion-free. We discuss the various possible definitions of L 2-signatures (using the signature operator, using the cap product of differential forms, using a cap product in cellular L 2-cohomology,...) in this situation, and prove that they all coincide.

In 1976, Atiyah [2] constructed the L2-Betti numbers of a compact Riemannian manifold. They are defined in terms of the spectrum of the Laplace operator on the universal covering of M. ByAtiyah’sL2-index theorem [2], they can be used e.g. to compute the Euler characteristic of M.

Abstract. In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups. In particular we describe the asymptotic behaviour of the integrated density of states (spectral distribution function) of these random Hamiltonians near the spectral minimum. The first part of the note discusses various aspects of the quantum percolation model, subsequently we formulate a series of new results, and finally we outline the strategy used to prove our main theorem. 1.

Abstract. In this paper, we prove that the L 2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, under some hypotheses. We also prove that some L 2 spectral invariants can be approximated by the corresponding average spectral invariants of a regular exhaustion. The main tool which is used is a generalisation of the ”principle of not feeling the boundary ” (due to M. Kac), for heat kernels associated to boundary value problems.

Abstract. We give results on the following questions about a topologically tame hyperbolic 3-manifold M: 1. Does M have nonzero L 2-harmonic 1-forms? 2. Does zero lie in the spectrum of the Laplacian acting on Λ 1 (M)/Ker(d)? 1.

Abstract not found

Given a C ∗-algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [26], and show that A R is a C ∗-algebra, and τ extends to a semicontinuous semifinite trace on A R. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A ′′ and can be approximated in measure by operators in A R, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a τ-a.e. bimodule on A R, denoted by A R, and such bimodule contains the functional calculi of selfadjoint elements of A R under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on A R.

Abstract. In this paper, we prove that the L 2 combinatorial torsion of an amenable covering space can be approximated by the combinatorial torsions of a regular exhaustion. An ancillary theorem shows the L 2 spectral density function of the combinatorial Laplacian on L 2-cochains on the covering space is approximated by the average spectral density functions of the combinatorial Laplacian on the cochains of the regular exhaustion, with either Dirichlet or Neumann boundary conditions, extending one of the main results in [DM]. The technique used incorporates some results of algebraic number theory.

Abstract not found