On some stable representations of hyperbolic groups

Abstract

Let $Γ$ be a hyperbolic group and G be the isometry group of a Gromov-hyperbolic, properand geodesic metric space. We study the action of the outer automorphism group Out($Γ$) onthe set X($Γ$,G) of conjugacy classes of representations of $Γ$ into G. We construct a familyof Out($Γ$)-invariant subsets of X($Γ$,G) which contains (stricly or not) the set of conjugacyclasses of quasi-convex representations and give a sufficient condition for the induced actionto be properly discontinuous. Finally, we give a criterion for a representation to have discreteimage and finite kernel and use it when G = Isom+(H3) to find new characterizations ofquasi-convex (i.e. convex cocompact) subgroups of PSL2(C).