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 and fundamental group G; (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 non-trivial conjugacy class of G which is finite. 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. 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. We also obtain precise information about the eta-invariants in question themselves, which are usually much more subtle objects than the rho-invariants.