# Bordism, rho-invariants and the Baum–Connes conjecture

### Paolo Piazza

Università di Roma La Sapienza, Italy### Thomas Schick

Georg-August-Universität Göttingen, Germany

## Abstract

Let $Γ$ 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 and fundamental group $Γ$;

(ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group $Γ$.
The invariants we consider are more precisely

- the Atiyah–Patodi–Singer (≡ APS) rho-invariant associated to a pair of finite dimensional unitary representations $λ_{1},λ_{2}:Γ→U(d)$,
- the $L_{2}$-rho-invariant of Cheeger–Gromov,
- the delocalized eta-invariant of Lott for a non-trivial conjugacy class of $Γ$ which is finite.

We prove that all these rho-invariants vanish if the group $Γ$ is *torsion-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 $A/[A,A]$ for 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.

## Cite this article

Paolo Piazza, Thomas Schick, Bordism, rho-invariants and the Baum–Connes conjecture. J. Noncommut. Geom. 1 (2007), no. 1, pp. 27–111

DOI 10.4171/JNCG/2