Teichmuller spaces for pointed Fuchsian groups and their modular groups

It is known that the modular group Mod(X) acts discontinuously (but not freely) on the Teichmuller space T(X) for a finite type Riemann surface X, while it does not necessarily act discontinuously on T(X) when X is of infinite type. The primary purpose of the paper is to discuss those subgroups of Mod(X) acting discontinuously and freely on T(X) and to discuss the properties of the corresponding quotient complex manifolds as well. Actually, we will discuss some generalized Teichmuller spaces, the Teichmuller 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]).