Wolfgang Lück

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.

The limit of _{}-Betti numbers of a tower of finite covers with amenable fundamental 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.

Twisting L2-invariants with finite-dimensional representations

We investigate how one can twist L^2-invariants such as L^2-Betti numbers and L^2-torsion with finite-dimensional representations. As a special case we assign to the universal covering of a finite connected CW-complex X together with an element phi in H^1(X;R) a phi-twisted L^2-torsion function from R^{>0} to R, provided that the fundamental group of X is residually finite and its universal covering is L^2-acyclic.

On the growth of Betti numbers in

We study the asymptotic growth of Betti numbers in tower of finite covers and provide simple proofs of approximation results, which were previously obtained by Calegari-Emerton, in the generality of arbitrary p-adic analytic towers of covers. Further, we also obtain partial results about arbitrary pro-

p

p

-adic analytic towers

towers.

L2-Euler characteristics and the Thurston norm: L2-EULER CHARACTERISTICS AND THE THURSTON NORM

We assign to a finite

TOPOLOGICAL K-THEORY OF THE GROUP C*-ALGEBRA OF A SEMI-DIRECT PRODUCT ℤn ⋊ ℤ/m FOR A FREE CONJUGATION ACTION

CW

L2-TORSION, THE MEASURE-THEORETIC DETERMINANT CONJECTURE, AND UNIFORM MEASURE EQUIVALENCE

-complex and an element in its first cohomology group a twisted version of the

Groups of small homological dimension and the Atiyah conjecture

L^2