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arXiv · 1208.0856

Finitely summable Fredholm modules for boundary actions of hyperbolic groups

Heath Emerson·Bogdan Nica
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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.

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Heath Emerson, Bogdan Nica. 2012-08-03. Finitely summable Fredholm modules for boundary actions of hyperbolic groups. https://arxiv.org/abs/1208.0856

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