Bordism, rho-invariants and the Baum–Connes conjecture

Abstract

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

• the Atiyah–Patodi–Singer (≡ APS) rho-invariant associated to a pair of finite dimensional unitary representations ${\lambda }_1,{\lambda }_2:\Gamma \to U(d)$,
• the $L^2$-rho-invariant of Cheeger–Gromov,
• the delocalized eta-invariant of Lott for a non-trivial conjugacy class of $\Gamma$ which is finite.

We prove that all these rho-invariants vanish if the group $\Gamma$ is torsion-free and the Baum–Connes map for the maximal group ${\mathrm{C}}^{\ast }$-algebra is bijective. This condition is satisfied, for example, by torsion-free amenable groups or by torsion-free discrete subgroups of $\mathrm{SO}(n,1)$ and $\mathrm{SU}(n,1)$. For the delocalized invariant we only assume the validity of the Baum–Connes conjecture for the reduced ${\mathrm{C}}^{\ast }$-algebra. In addition to the examples above, this condition is satisfied e.g. by Gromov hyperbolic groups or by cocompact discrete subgroups of $\mathrm{SL}(3,C)$.

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 ${\mathrm{C}}^{\ast }$-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.