Equivalence of primitive-stable and Bowditch actions of the free group of rank two on Gromov-hyperbolic spaces

TL;DR

This paper proves that for the free group of rank two acting on Gromov-hyperbolic spaces, the sets of Bowditch and primitive-stable representations are identical, extending previous results and providing an independent proof in the case of PSL(2,C).

Contribution

The paper establishes the equivalence of Bowditch and primitive-stable representations for rank two free groups on Gromov-hyperbolic spaces, offering a new independent proof for the PSL(2,C) case.

Findings

Bowditch and primitive-stable representations are equal for rank two free groups.

The proof extends previous results and is independent of earlier proofs.

The equivalence holds specifically in the context of Gromov-hyperbolic spaces.

Abstract

We prove that the set of Bowditch representations (introduced by Bowditch in 1998, then generalized by Tan, Wong and Zhang in 2008) and the set of primitive-stable representations (introduced by Minsky in 2013) of the free group of rank two in the isometry group of a Gromov-hyperbolic space are equal. The case of $\mathrm{PSL}(2,\mathbb{C})$-representations has already been proved by Lee and Xu and independently Series. Our proof in this context is independent.