A note on the higher Atiyah-Patodi-Singer index theorem on Galois coverings

Let G be a finitely generated discrete group satisfying the rapid decay condition. We give a new proof of the higher Atiyah-Patodi-Singer theorem on a Galois G-coverings, thus providing an explicit formula for the higher index associated to a group cocycle which is of polynomial growth with respect to a word-metric. Our new proof employs relative K-theory and relative cyclic cohomology in an essential way.