Pseudo-Anosov flows and the geometry of Anosov-like group actions

TL;DR

This paper demonstrates that pseudo-Anosov flows induce isometric actions on Gromov-hyperbolic spaces and explores the geometric group-theoretic properties of these actions on bifoliated planes.

Contribution

It establishes a connection between pseudo-Anosov flows and hyperbolic geometry, revealing new properties of group actions and generic elements in the fundamental group.

Findings

Flow actions can be viewed as isometric actions on Gromov-hyperbolic spaces.

Non-ℝ-covered flows admit elements with weak proper discontinuity.

Elements not representing periodic orbits are generic in the fundamental group.

Abstract

We show that the action on its orbit space induced by a pseudo-Anosov flow on a closed $3$-manifold (and more general Anosov-like actions) can be seen as an isometric action on a Gromov-hyperbolic space. When the flow is not $\R$-covered, we show that this action admits elements that are weakly properly discontinuous and deduce that elements of $\pi_1(M)$ that do \emph{not} represent a periodic orbit of the flow are generic for any word metric coming from a finite generating set. We also give a number of other geometric group-theoretic results for Anosov-like group actions on bifoliated planes.