A geometric correspondence for reparameterizations of geodesic flows

TL;DR

This paper establishes a geometric correspondence between hyperbolic group metrics and flow reparameterizations, producing new examples of geodesic flows with integer-length periodic orbits and analyzing their geometric properties.

Contribution

It introduces a novel correspondence linking hyperbolic metrics and flow reparameterizations, and constructs new examples of geodesic flows with specific periodic orbit properties.

Findings

First examples of continuous reparameterizations with integer-length periodic orbits

Isometric actions on Gromov-hyperbolic spaces with non-simple elements as loxodromic

Continuity of the Bowen--Margulis--Sullivan geodesic current map on moduli space

Abstract

For any non-elementary, torsion-free hyperbolic group, we provide a correspondence between the left-invariant Gromov-hyperbolic metrics on the group that are quasi-isometric to a word metric, and continuous reparameterizations of the associated Mineyev's flow space. From this correspondence, we produce the first examples of continuous reparameterizations of geodesic flows on negatively curved manifolds with all periodic orbits having integer lengths. For surface and free groups, this also yields isometric actions on Gromov-hyperbolic spaces on which loxodromic elements are precisely the non-simple elements. Key ingredients in our proof are an analysis of the geometry of Mineyev's flow space (such as the metric-Anosov property recently proven by Dilsavor), and the density of Green metrics in the moduli space of (symmetric) metrics on the group. We further establish continuity of the Bowen--Margulis--Sullivan geodesic current map on the moduli space of metrics, as well as a Bowen-type description of these currents as limits of sums of appropriately normalized atomic geodesic currents. For surface groups, we apply this continuity result to show that the Bowen--Margulis--Sullivan map restricts to a topological embedding on Hitchin components (up to contragradient involution) when equipped with their Hilbert lengths.