  Let $\Gamma$ 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 $\Gamma$-coverings, thus providing an explicit formula for the
higher index associated to a group cocycle $c\in Z^k (\Gamma;\mathbb{C})$ 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.
