Thomas Schick

Approximating L2‐invariants and the Atiyah conjecture

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 that are residually torsion-free elementary amenable or are residually free. This result implies that there are no nontrivial zero divisors in ℂG. The statement relies on new approximation results for L2-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 L2-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 . 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 .

On a question of Atiyah

In this note we explain how the computation of the spectrum of the lamplighter group from \cite{Grigorchuk-Zuk(2000)} yields a counterexample to a strong version of the Atiyah conjectures about the range of

ON THE TOPOLOGY OF T-DUALITY

L^2

On the Equivalence of Geometric and Analytic K-Homology

-Betti numbers of closed manifolds.Abstract In this Note we explain how the computation of the spectrum of the lamplighter group from [9] yields a counterexample to a strong version of the Atiyah conjectures about the range of L 2 -Betti numbers of closed manifolds.

The Spectral Measure of Certain Elements of the Complex Group Ring of a Wreath Product

We study a topological version of the T-duality relation between pairs consisting of a principal U(1)-bundle equipped with a degree-three integral cohomology class. We describe the homotopy type of a classifying space for such pairs and show that it admits a selfmap which implements a T-duality transformation. We give a simple derivation of a T-duality isomorphism for certain twisted cohomology theories. We conclude with some explicit computations of twisted K-theory groups and discuss an example of iterated T-duality for higher-dimensional torus bundles.

Bordism, rho-invariants and the Baum-Connes conjecture

We give a proof that the geometric K-homology theory for finite CWcomplexes defined by Baum and Douglas is isomorphic to Kasparov’s Khomology. The proof is a simplification of more elaborate arguments which deal with the geometric formulation of equivariantK-homology theory.

Enlargeability and index theory

We use elementary methods to compute the L2-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk and Zuk, and later used by them, together with Linnell and Schick, to produce a counterexample to a strong version of the Atiyah conjecture about the range of L2-Betti numbers. We use our results to construct manifolds with certain L2-Betti numbers (given as convergent infinite sums of rational numbers) which are not obviously rational, but we have been unable to determine whether any of them are irrational.

Integrality of L 2 -Betti numbers

Letbe a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to (i) the spin Dirac operator of a spin manifold with positive scalar curvature and fundamental group� ; (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group� . The invariants we consider are more precisely � the Atiyah-Patodi-Singer (� APS) rho-invariant associated to a pair of finite dimensional unitary representations� 1;� 2W� ! U.d/, � theL 2 -rho-invariant of Cheeger-Gromov, � the delocalized eta-invariant of Lott for a non-trivial conjugacy class ofwhich is finite. We prove that all these rho-invariants vanish if the groupistorsion-free and the Baum-Connes map for the maximal group C*-algebra is bijective. This condition is satisfied, for example, by torsion-free amenable groups or by torsion-free discrete subgroups of SO.n;1/ and SU.n;1/. For the delocalized invariant we only assume the validity of the Baum-Connes conjecture for the reduced C*-algebra. In addition to the examples above, this condition is satisfied e.g. by Gromov hyperbolic groups or by cocompact discrete subgroups of SL.3; C/. In particular, the three rho-invariants associated to the signature operator are, for such groups, homotopy invariant. For the APS and the Cheeger-Gromov rho-invariants the latter result had been established by Navin Keswani. Our proof reestablishes this result and also extends it to the delocalized eta-invariant of Lott. The proof exploits in a fundamental way results from bordism theory as well as various generalizations of the APS-index theorem; it also embeds these results in general vanishing phenomena for degree zero higher rho-invariants (taking values in A=ŒA;Afor suitable C*-algebras A). We also obtain precise information about the eta-invariants in question themselves, which are usually much more subtle objects than the rho-invariants.

Finite group extensions and the Atiyah conjecture

Let M be a closed enlargeable spin manifold. We show non-triviality of the universal index obstruction in the K-theory of the maximal

The space of metrics of positive scalar curvature

C^*