Amalgam Anosov representations

TL;DR

This paper introduces amalgam Anosov representations for hyperbolic groups, establishing their properties and demonstrating they form a domain of discontinuity under the outer automorphism group action.

Contribution

It defines amalgam Anosov representations using JSJ splittings and proves their domain of discontinuity, extending the understanding of Anosov representations in hyperbolic groups.

Findings

Amalgam Anosov representations form a domain of discontinuity for Out(Γ).

Representation restrictions to vertex groups determine amalgam Anosov status.

The paper links Anosov properties of vertex groups to the global representation.

Abstract

Let $\Gamma$ be a one-ended, torsion-free hyperbolic group and let $G$ be a semisimple Lie group with finite center. Using the canonical JSJ splitting due to Sela, we define amalgam Anosov representations of $\Gamma$ into $G$ and prove that they form a domain of discontinuity for the action of $\mathrm{Out}(\Gamma)$. In the appendix, we prove, using projective Anosov Schottky groups, that if the restriction of the representation to every Fuchsian or rigid vertex group of the JSJ splitting of $\Gamma$ is Anosov, with respect to a fixed pair of opposite parabolic subgroups, then $\rho$ is amalgam Anosov.