Some New Directions in Teichmueller Theory

Abstract

In this thesis we will extend the study of Teichmueller spaces in two relatively unexplored new directions. First, beginning with the Teichmueller space of the flat 2-torus, rather than increasing the genus, we will explore higher dimensional tori. This yields Riemannian symmetric spaces with very different, yet analogous, behavior to classically studied Teichmueller spaces of hyperbolic surfaces. Second, in the setting of hyperbolic surfaces, we study a certain kind of rigidity for maps between different Teichmueller spaces. We will classify most of the possible cases of holomorphic isometric submersions between Teichmueller spaces of finite-type hyperbolic surfaces and begin exploration in the case of infinite-type.