Higher Atiyah-Patodi-Singer index theory on Galois coverings

We consider a Galois covering with boundary Gamma --> (M) over tilde --> M 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. We explain how it is possible to develop a higher Atiyah-Patodi-Singer index theory by employing the notion of noncommutative spectral section and the b-calculus on the covering (M) over tilde. Our main theorem extends the higher index theorem of Connes-Moscovici to such Gamma-Galois coverings with boundary.