Bordism, rho-invariants and the Baum–Connes conjecture

Let G be a finitely generated discrete group. In this paper we establishvanishing results for rho-invariants associated to (i) the spin-Dirac operator of a spin manifold with positive scalar curvature (ii) the signature operator of the disjoint union of a pair of homotopyequivalent oriented manifolds with fundamental group G. The invariants we consider are more precisely - the Atiyah-Patodi-Singer rho-invariant associated to a pair of finitedimensional unitary representations. - the L2-rho invariant of Cheeger-Gromov - the delocalized eta invariant of Lott for a finite conjugacy class of G. We prove that all these rho-invariants vanish if the group G is torsion-freeand the Baum-Connes map for the maximal group C^*-algebra is bijective. For thedelocalized invariant we only assume the validity of the Baum-Connes conjecturefor the reduced C^*-algebra. In particular, the three rho-invariants associated to the signature operatorare, for such groups, homotopy invariant. For the APS and the Cheeger-Gromovrho-invariants the latter result had been established by Navin Keswani. Ourproof re-establishes this result and also extends it to the delocalizedeta-invariant of Lott. Our method also gives some information about theeta-invariant itself (a much more saddle object than the rho-invariant).