The trace on the K-theory of group C*-algebras

The canonical trace on the reduced C*-algebra of a discrete group gives rise to a homomorphism from the K-theory of this C^*-algebra to the real numbers. This paper addresses the range of this homomorphism. For torsion free groups, the Baum-Connes conjecture and Atiyah's L2-index theorem implies that the range consists of the integers. If the group is not torsion free, Baum and Connes conjecture that the trace takes values in the rational numbers. We give a direct and elementary proof that if G acts on a tree and admits a homomorphism \alpha to another group H whose restriction to every stabilizer group of a vertex is injective, then the range of the trace for G, tr_G(K(C_r^*G)) is contained in the range of the trace for H, tr_H(K(C_r^*H)). This follows from a general relative Fredholm module technique. Examples are in particular HNN-extensions of H where the stable letter acts by conjugation with an element of H, or amalgamated free products G=H*_U H of two copies of the same groups along a subgroup U.Comment: Reference added. AMSLateX2e, 12 pages. Preprint-Series SFB Muenster, No 66, to appear in Duke Math. Journa

Bordism, rho-invariants and the Baum-Connes conjecture

Let G be 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 (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group G. The invariants we consider are more precisely - the Atiyah-Patodi-Singer rho-invariant associated to a pair of finite dimensional unitary representations. - the L2-rho invariant of Cheeger-Gromov - the delocalized eta invariant of Lott for a finite conjugacy class of G. We prove that all these rho-invariants vanish if the group G is torsion-free and the Baum-Connes map for the maximal group C^*-algebra is bijective. For the delocalized invariant we only assume the validity of the Baum-Connes conjecture for the reduced C^*-algebra. 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 re-establishes this result and also extends it to the delocalized eta-invariant of Lott. Our method also gives some information about the eta-invariant itself (a much more saddle object than the rho-invariant).Comment: LaTeX2e, 60 pages; the gap pointed out by Nigel Higson and John Roe is now closed and all statements of the first version of the paper are proved (with some small refinements

L2-determinant class and approximation of L2-Betti numbers

A standing conjecture in L2-cohomology is that every finite CW-complex X is of L2-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L2-acyclic, we also prove that the L2-determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation formulas for L2-Betti numbers. Errata are added, rectifying some unproved statements about "amenable extension": throughout, amenable extensions should be extensions with \emph{normal} subgroups.Comment: amsLaTeX2e, 26 pages; v2: Errata are added, rectifying some unproved statements about "amenable extension