Secondary higher invariants and cyclic cohomology for groups of polynomial growth

Abstract

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