A lower bound for Torelli-K-quasiconformal homogeneity

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.

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