Rigidity for quasi-Fuchsian actions on negatively curved spaces

We prove that if $G\curvearrowright X$ is a convex cocompact isometric group action on a CAT(−1) space, and the limit set has Hausdorff and topological dimensions equal to 1, then the action preserves a convex subset Y⊆X isometric to ℍ². This implies a conjecture by M. Bourdon. Our methods also yield a new proof of the n-dimensional analog of this statement for n≥².