Secondary higher invariants and cyclic cohomology for groups of polynomial growth

We prove that if $\Gamma$ is a group of polynomial growth, then each delocalized cyclic cocycle on the algebra has representative growth. For cocycle, we thus define higher analogue Lott’s eta invariant and its convergence for invertible differential operators. also use determinant map construction Xie Yu to there well-defined pairing between cocycles $K$-theory classes $C^\*$-algebraic secondary invariants. When this class rho an operator, show precisely aforementioned invariant. As application equivalence, provide Atiyah–Patodi–Singer index theorem, given $M$ compact spin manifold with boundary, equipped positive scalar metric $g$ having fundamental $\Gamma=\pi\_1(M)$ which finitely generated