Commutator-stable representations of hyperbolic groups

Remfort-Aurat, Ulysse

doi:10.2140/agt.2026.26.2559

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Commutator-stable representations of hyperbolic groups

Ulysse Remfort-Aurat

Algebraic & Geometric Topology 26 (2026) 2559–2588

DOI: 10.2140/agt.2026.26.2559

Abstract

Let $Γ$ be a hyperbolic group and $G$ be the isometry group of a Gromov-hyperbolic and geodesic metric space. We study the action of the outer automorphism group $Out (Γ)$ on the set $𝒳 (Γ, G)$ of conjugacy classes of representations of $Γ$ into $G$ . For some $Aut (Γ)$ -invariant subset $A \subset Γ$ , we study the properties of $A$ -stable representations which are generalizations of primitive stable representations defined by Minsky. Their conjugacy classes form $Out (Γ)$ -invariant subsets of $𝒳 (Γ, G)$ which contains the set of conjugacy classes of quasi-isometrically embedded representations. We establish a sufficient condition for the induced action to be properly discontinuous and use it when $A$ is the set of commutators of $Γ$ . Working with the derived subgroup $[Γ, Γ]$ , we also prove that $[Γ, Γ]$ -stability is equivalent to the $[Γ, Γ]$ -well-displacing property. Focusing on representations in ${PSL}_{2} (ℂ)$ , we find new characterizations of convex cocompact subgroups of ${PSL}_{2} (ℂ)$ .

Keywords

hyperbolic group, convex cocompact Kleinian groups, real tree, properly discontinuous action

Mathematical Subject Classification

Primary: 30L05, 57M60

Secondary: 20E08, 20F65

References

Publication

Received: 6 September 2024

Revised: 17 June 2025

Accepted: 7 July 2025

Published: 1 September 2026

Authors

Ulysse Remfort-Aurat
Mathematics Department Aix-Marseille Université, CNRS, I2M Marseille France

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Volume 26, issue 7 (2026)