Groups of QC homeomorphisms on Riemann surfaces
by
Tatsuhiko Yagasaki
Kyoto Institute of Technology

We investigate the topological types of groups of quasiconformal (QC) homeomorphisms on Riemann surfaces with the compact-open topology. Suppose M is a connected Riemann surface. Let H(M) denote the homeomorphism group of M and H^(L)QC(M) denote the subgroup of (locally) QC homeomorphisms of M. The superscript + means orientation preserving and 0 denotes the identity connected component of the corresponding group.

Main Theorem
(i) If M is compact, then (H(M)₊, H^QC(M)) is an (s, Sigma)-manifold.
(ii) If M is noncompact, then (H(M)₀, H^LQC(M)₀, H^QC(M)₀) is an (s^infinity, Sigma^infinity, Sigma^infinity_f)-manifolds.

Corollary
(H(M)₀, H^QC(M)₀) is homeomorphic to the product of (s, Sigma) and N, where N is defined as follows:
Compact case : g = the genus of M.
(i) If g = 0 then N = SO(3), (ii) If g = 1 then N = the torus, (iii) If g >= 2 then N = 1 point.
Noncompact case :
(i) If M = R² or R^2 - 1 pt, then N = the circle, (ii) otherwise N = 1 pt.

Date received: July 22, 1999