The space of metric structures on hyperbolic groups

TL;DR

This paper investigates the structure and properties of the space of hyperbolic pseudometrics on hyperbolic groups, revealing its geometric features, group actions, and connections to geodesic currents, extending known results from surface and free groups.

Contribution

It introduces a natural metric on the space of hyperbolic pseudometrics, proves its contractibility and properness properties, and extends the Bowen-Margulis map to this setting.

Findings

The space of hyperbolic pseudometrics is unbounded and contractible.

The outer automorphism group acts properly by isometries.

Continuity results for the Bowen-Margulis map and mean distortion are established.

Abstract

We study the metric and topological properties of the space $\mathscr{D}(G)$ of left-invariant hyperbolic pseudometrics on the non-elementary hyperbolic group $G$ that are quasi-isometric to a word metric, up to rough similarity. This space naturally contains the Teichm\"uller space in case $G$ is a surface group and the Culler-Vogtmann outer space when $G$ is a free group. Endowed with a natural metric reminiscent of the (symmetrized) Thurston's metric on Teichm\"uller space, we prove that $\mathscr{D}(G)$ is an unbounded contractible metric space and that $\mathrm{Out}(G)$ acts metrically properly by isometries on it. If we restrict ourselves to the subspace $\mathscr{D}_{\delta}(G)$ of the points represented by $\delta$-hyperbolic metrics with critical exponent 1, we prove that it is either empty or proper. We also prove continuity of the Bowen-Margulis map from $\mathscr{D}_{\delta}(G)$ into the space $\mathbb{P}\mathcal{C}urr(G)$ of projective geodesic currents on $G$, extending similar results for surface and free groups, and the continuity of the (normalized) mean distortion as a function on $\mathscr{D}(G)\times \mathscr{D}(G)$.