On purely loxodromic actions

TL;DR

This paper constructs a specific hyperbolic graph action of the free group F(a,b) that is acylindrical and purely loxodromic, with non-zero translation lengths and quasiconvex orbits, but the orbit map is not a quasi-isometric embedding.

Contribution

It provides a novel example of a hyperbolic group action with properties that challenge previous assumptions about orbit maps and quasi-isometric embeddings.

Findings

Constructed a hyperbolic graph action with specified properties.

Showed the orbit map is not a quasi-isometric embedding despite other hyperbolic features.

Demonstrated the existence of purely loxodromic acylindrical actions with non-zero translation lengths.

Abstract

We construct an example of an isometric action of $F(a,b)$ on a $\delta$-hyperbolic graph $Y$, such that this action is acylindrical, purely loxodromic, has asymptotic translation lengths of nontrivial elements of $F(a,b)$ separated away from $0$, has quasiconvex orbits in $Y$, but such that the orbit map $F(a,b)\to Y$ is not a quasi-isometric embedding.