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Two applications of strong hyperbolicity. (English) Zbl 1470.20020

Summary: We present two analytic applications of the fact that a hyperbolic group can be endowed with a strongly hyperbolic metric. The first application concerns the crossed product \(C^\ast\)-algebra defined by the action of a hyperbolic group on its boundary. We construct a natural time flow, involving the Busemann cocycle on the boundary. This flow has a natural KMS state, coming from the Hausdorff measure on the boundary, which is furthermore unique when the group is torsion-free. The second application is a short new proof of thxe fact that a hyperbolic group admits a proper isometric action on an \(\ell^p\)-space for large enough \(p\).

MSC:

20F67 Hyperbolic groups and nonpositively curved groups
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