The space of metric structures on hyperbolic groups
Eduardo Oreg\'on-Reyes

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 of left-invariant hyperbolic pseudometrics on the non-elementary hyperbolic group that are quasi-isometric to a word metric, up to rough similarity. This space naturally contains the Teichm\"uller space in case is a surface group and the Culler-Vogtmann outer space when is a free group. Endowed with a natural metric reminiscent of the (symmetrized) Thurston's metric on Teichm\"uller space, we prove that is an unbounded contractible metric space and that acts metrically properly by isometries on it. If we restrict ourselves to the subspace of the points represented by -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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Geometric Analysis and Curvature Flows · Mathematical Dynamics and Fractals
