# Two applications of strong hyperbolicity

## TL;DR

This paper explores two analytic applications of strongly hyperbolic metrics on hyperbolic groups, including constructing a natural time flow with a unique KMS state and providing a new proof of proper isometric actions on ℓ^p-spaces.

## Contribution

It introduces novel applications of strong hyperbolicity, including a natural time flow with a unique KMS state and a simplified proof of proper isometric actions on ℓ^p-spaces.

## Key findings

- Construction of a natural time flow with a KMS state from the boundary measure.
- Uniqueness of the KMS state for torsion-free hyperbolic groups.
- A new proof that hyperbolic groups act properly on ℓ^p-spaces for large p.

## Abstract

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*-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 the fact that a hyperbolic group admits a proper isometric action on an $\ell^p$-space, for large enough $p$.