Finitely summable Fredholm modules for boundary actions of hyperbolic groups

TL;DR

This paper constructs finitely summable Fredholm modules for hyperbolic group boundary actions, linking their summability to boundary dimension and computing the Connes-Chern character in cyclic cohomology.

Contribution

It introduces a family of Fredholm modules over boundary crossed product C*-algebras that represent the boundary extension class in K-homology, with summability tied to boundary Hausdorff dimension.

Findings

Fredholm modules represent the boundary extension class in K-homology.

Summability relates to the Hausdorff dimension of the boundary.

Connes-Chern character of the boundary extension is explicitly computed.

Abstract

We construct a family of odd, finitely summable Fredholm modules over the crossed product C*-algebra $C(\bd \G)\rtimes \G$ associated to the action of a non-elementary hyperbolic group $\G$ on its Gromov boundary $\bd \G$. These Fredholm modules all represent the same, distinguished class in K-homology, namely that of the `boundary extension' of $C(\bd \G)\rtimes \G$ associated to the Gromov compactification of $\G$, and is typically nonzero. Their summability is closely related to the Hausdorff dimension of the boundary. We use these results to compute the Connes-Chern character of the boundary extension in cyclic cohomology.