Bordism, rho-invariants and the Baum–Connes conjecture

Let ! 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 !; (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group ! . The invariants we consider are more precisely ! theAtiyah–Patodi–Singer (!APS) rho-invariant associated to a pair of finite dimensional unitary representations "1;"2 W ! ! U.d/, ! the L-rho-invariant of Cheeger–Gromov, ! the delocalized eta-invariant of Lott for a non-trivial conjugacy class of ! which is finite. We prove that all these rho-invariants vanish if the group! 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=ŒA;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. Mathematics Subject Classification (2000). 58J28, 19K56.