Manhattan geodesics and the boundary of the space of metric structures on hyperbolic groups

Abstract

For any non-elementary hyperbolic group $\Gamma$, we find an outer automorphism invariant geodesic bicombing for the space of pseudometric structures on $\Gamma$ equipped with a symmetrized version of the Thurston metric on Teichmüller space. We construct and study a boundary for this space and show that it contains many well-known pseudometrics. As corollaries we obtain results regarding continuous extensions of translation length functions to the space of geodesic currents and settle a conjecture of Bonahon in the negative.