Arithmeticity in low dimensional geometry

Abstract

The importance of manifolds with constant sectional curvatures equal to -1 can hardly be overstated. Some of the milestones in modern Mathematics, such as Riemann's Uniformisation Theorem and Thurston's Hyperbolisation Theorem, revolve around hyperbolic 2- and 3-manifolds. These objects, in turn, are inexorably linked to the understanding of Fuchsian and Kleinian groups.The arithmetic properties of Fuchsian and Kleinian groups have been a successful topic of research pertaining to a variety of fields, from Geometric Topology to Dynamics, from Number Theory to Theoretical Physics. In a tradition spanning more than 50 years, mathematicians have studied these properties, and used them, in order to obtain new information on the geometry and spectrum of associated hyperbolic manifolds.This research proposal is primarily concerned with a specific class of hyperbolic surfaces, known as semi-arithmetic surfaces. While still carrying rich arithmetic data, these surfaces are surprisingly abundant in deformation spaces of hyperbolic structures. We plan to investigate different aspects of semi-arithmetic surfaces, and to explore how their geometrical properties may interact with algebraic and dynamical properties. (AU)