Bordism, rho-invariants and the Baum–Connes conjecture

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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 We prove that all these rho-invariants vanish if the group Γ is torsion-free and the Baum–Connes map for the maximal group \mathrm C^* -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^* -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^* -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.

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