2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123050We strengthen the analogy between convex co-compact Kleinian groups and convex co-compact subgroups of the mapping class group of a surface (in the sense of B. Farb and L. Mosher).v3. Final version accepted for publication. Minor revisions, updated references. v2. Theorems 3.9 and 3.10 strengthened, proof of Theorem 5.4 shortened, and questions in Section 8 modified. References to Brock--Farb, Duchin, and Hamenstadt addedGeometric TopologyGroup Theory20F65 (Primary) 30F60, 57M07, 57M50 (Secondary)text