Homotopy invariance of twisted higher signatures on manifolds with boundary

Let M be an oriented compact manifold with boundary. We assume that pi(1) (M) is the product of a non-trivial finite group F and of a group F which is either of polynomial growth or Gromov hyperbolic. We fix a non-trivial representation rho:F --> U(l) and let E-rho be the associated unitary flat bundle on M. We denote by M the universal cover of M and we consider the Gamma-Galois covering pi: M/F --> M, the lifted flat bundle E-rho = pi*(E-rho) and the associated twisted signature operator. Under the additional assumption that the induced twisted signature operator on the boundary of M/F is L-2-invertible, Lott has introduced in [L2] the twisted higher signatures of M; our main result, a positive answer to a form of Novikov conjecture on manifolds with boundary, is that these are homotopy invariants of the pair (M, partial derivativeM). The proof depends heavily on the b-beta(infinity)-pseudodifferential calculus developed in [LP1], on the higher APS index theorem of [LP1] (here extended so as to cover Gromov-hyperbolic groups) and on a classical result of Kaminker-Miller, stating the equality of the index classes associated to two homotopy-equivalent hermitian Fredholm complexes.