The space of metric structures on hyperbolic groups

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 Teichmüller 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 Teichmüller 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 $\mathscr{D}_δ(G)$ of the points represented by $δ$-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 $\mathscr{D}_δ(G)$ 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 $\mathscr{D}(G)\times \mathscr{D}(G)$.