The groups of quasiconformal homeomorphisms on Riemann surfaces

Abstract:

Suppose $M$ is a connected Riemann surface. Let ${\mathcal H}(M)$ denote the homeomorphism group of $M$ with the compact-open topology, and ${\mathcal H}^{\mathrm {QC}}(M)$ denote the subgroup of quasiconformal mappings of $M$ onto itself, and let ${\mathcal H}(M)_0$ and ${\mathcal H}^{\mathrm {QC}}(M)_0$ denote the identity components of ${\mathcal H}(M)$ and ${\mathcal H}^{\mathrm {QC}}(M)$ respectively. In this paper we show that the pair $({\mathcal H}(M)_0, {\mathcal H}^{\mathrm {QC}}(M)_0)$ is an $(s, \Sigma )$-manifold, and determine their topological types.