  A closed hyperbolic Riemann surface M is said to be K-quasiconformally
homogeneous if there exists a transitive family F of K-quasiconformal
homeomorphisms. Further, if all [f] in F act trivially on H1(M;Z), we say M is
Torelli-K-quasiconformally homogeneous. We prove the existence of a uniform
lower bound on K for Torelli-K-quasiconformally homogeneous Riemann surfaces.
This is a special case of the open problem of the existence of a lower bound on
K for (in general non-Torelli) K-quasiconformally homogeneous Riemann surfaces.
