  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
  (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 finite conjugacy class of G.
  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. For the
delocalized invariant we only assume the validity of the Baum-Connes conjecture
for the reduced C^*-algebra.
  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 re-establishes this result and also extends it to the delocalized
eta-invariant of Lott. Our method also gives some information about the
eta-invariant itself (a much more saddle object than the rho-invariant).
