Teichmüller spaces of non-discrete groups

Abstract

The concept of Teichmüller space of a Fuchsian group can be extended to any non-discrete group of conformal homeomorphisms of the hyperbolic plane. In this paper, we first present three models of Teichmüller space fulfilling that goal. Then we use them to study the Teichmüller spaces T(G) of the non-discrete subgroups G of PSL(2,R). We show that T(G) is not trivial if and only if G is a subgroup consisting of hyperbolic elements with two common fixed points and accumulating to at least one non-identity element. Furthermore, we show that if T(G) is not trivial, then (1) T(G) is conformally equivalent to the open unit disk, (2) the Teichmüller metric on T(G) is equal to the hyperbolic metric on the disk, and (3) the length spectrum is just a pseudometric on T(G) and when restricted to a one-dimensional real slice, it is a metric coinciding with the Teichmüller metric.