Spectral sections and higher Atiyah-Patodi-Singer index theory on Galois coverings

In this paper we consider Gamma --> (M) over tilde --> M, a Galois covering with boundary and (D) over tilde a Gamma-invariant generalized Dirac operator on (M) over tilde. We assume that the group Gamma is of polynomial growth with respect to a word metric. By employing the notion of noncommutative spectral section associated to the boundary operator <(D)over tilde (0)> and the b-calculus on Galois coverings with boundary, we develop a higher Atiyah-Patodi-Singer index theory. Our main theorem extends to such Gamma-Galois coverings with boundary the higher index theorem of Connes-Moscovici.