Bordism, rho-invariants and the baum-connes conjecture

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 λ1, λ2 : 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. © 2013 EMS Publishing House.