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

TL;DR

This paper introduces a new boundary and geodesic structure for the space of metric structures on hyperbolic groups, linking various known metrics and deriving rigidity, regularity, and growth results.

Contribution

It constructs an invariant geodesic bicombing and boundary for the metric space of hyperbolic group structures, connecting multiple pseudo metrics and settling a conjecture.

Findings

Boundary contains metrics from actions on CAT(0) cube complexes, real trees, and coned-off Cayley graphs.

Establishes length spectrum rigidity and regularity of Manhattan curves.

Provides growth rate results for Anosov representations and extends translation distances to geodesic currents.

Abstract

For any non-elementary hyperbolic group $\Gamma$, we find an outer automorphism invariant geodesic bicombing for the space of metric structures on $\Gamma$ equipped with a symmetrized version of the Thurston metric on Techim\"uller space. We construct and study a boundary for this space and show that it contains many well-known pseudo metrics including those coming from actions on $\text{CAT}(0)$ cube complexes, real trees and coned-off Cayley graphs. As corollaries we deduce length spectrum rigidity results, regularity results for Manhattan curves, optimal growth rate results for Anosov representations and results regarding continuous extensions of translation distance functions to the space of geodesic currents. Using our results for geodesic currents we settle a conjecture of Bonahon in the negative.