Teichmüller spaces for pointed Fuchsian groups and their modular groups

Abstract

It is known that the modular group $\mathrm{Mod}\left(X\right)$ acts discontinuously (but not freely) on the Teichmüller space $T\left(X\right)$ for a finite type Riemann surface $X$, while it does not necessarily act discontinuously on $T\left(X\right)$ when $X$ is of infinite type. The primary purpose of the paper is to discuss those subgroups of $\mathrm{Mod}\left(X\right)$ acting discontinuously and freely on $T\left(X\right)$ and to discuss the properties of the corresponding quotient complex manifolds as well. Actually, we will discuss some generalized Teichmüller spaces, the Teichmüller spaces for pointed Riemann surfaces and pointed Fuchsian groups, and their modular groups, generalizing and completing some results of Bers [Be1], Kra [Kr1] and Nag ([Na1], [Na3], [Na4]).