Commutator-stable representations of hyperbolic groups

REMFORT-AURATLet 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./ on the set X .;G/ of conjugacy classes of representations of into G.For some Aut./-invariant subset A , we study the properties of A-stable representations which are generalizations of primitive stable representations defined by Minsky.Their conjugacy classes form Out./-invariant subsets of X .;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 .Working with the derived subgroup Œ; , we also prove that Œ; -stability is equivalent to the Œ;well-displacing property.Focusing on representations in PSL 2 .C/, we find new characterizations of convex cocompact subgroups of PSL 2 .C/. 1. Introduction 2559 2. Hyperbolic metric spaces 2563 3. Space of representations 2566 4. A-stable representations 2567 5. Actions on R-trees and testable subsets 2572 6.First examples 2576 7. New examples 2578 8. Representations in PSL 2 .C/ 2583 Appendix A. A finite testable subset for ŒF k ;