Note on C*-algebras associated to boundary actions of hyperbolic 3-manifold groups

TL;DR

This paper demonstrates that for certain hyperbolic 3-manifold groups, the isomorphism types of their associated crossed product C*-algebras depend solely on the manifold's homology, revealing infinitely many non-isomorphic groups with isomorphic C*-algebras.

Contribution

It applies Kirchberg-Phillips classification to relate C*-algebra isomorphism types to manifold homology, showing non-dynamical isomorphisms among hyperbolic groups.

Findings

Isomorphism types depend only on manifold homology

Infinitely many non-isomorphic groups have isomorphic crossed products

Isomorphisms are not induced by groupoid isomorphisms

Abstract

Using Kirchberg-Phillips' classification of purely infinite C*-algebras by K-theory, we prove that the isomorphism types of crossed product C*-algebras associated to certain hyperbolic 3-manifold groups acting on their Gromov boundary only depend on the manifold's homology. As a result, we obtain infinitely many pairwise non-isomorphic hyperbolic groups all of whose associated crossed products are isomorphic. These isomomorphisms are not of dynamical nature in the sense that they are not induced by isomorphisms of the underlying groupoids.