Linearization and mappings onto pseudocircle domains

Abstract

We demonstrate the existence of linearizations for groups of conformal and anticonformal homeomorphisms of Riemann surfaces. The finitely generated groups acting on plane domains are classified in terms of specific linearizations. This extends Maskit’s work in the directly conformal case. As an application we prove that there exist conformal representations of finite genus open Riemann surfaces for which accessible boundary points are either isolated or lie on circular arcs of pseudocircular boundary components. In many cases these are actually circle domains. Along the way we extend the applicability of Carathéodory’s boundary correspondence theorem for prime ends.