A classic geometry view of Teichmüller theory and variations on the Gromov-Lawson-Thurston conjecture

Abstract

This project aims to investigate the interplay between the following three areas of study: Teichmüller theory, dynamical systems, and real/complex hyperbolic geometry. Traditionally, Teichmüller's theory is based on complex analysis and hyperbolic geometry. One of the objectives of this project is to investigate Teichmuller theory from the viewpoint of classic geometry (an approach to hyperbolic geometry with a linear algebraic flavor on which the candidate is proficient), following the work of Sasha Anan'in and the ideas of the higher Teichmüller theory. The interplay between Teichmuller theory, dynamical systems, and hyperbolic 3-manifolds is well known from the work of Thurston, where the study of pseudo-Anosov maps is crucial, an area where the mentor for this project, André Salles de Carvalho, is an expert.On the other hand, the candidate is knowledgeable about the problems gravitating around the uniformization of four-dimensional manifolds with real and complex hyperbolic structures. Sometimes the dynamics of discrete isometry groups on the boundary of 4-dimensional complex hyperbolic spaces, which is a 3-sphere, produce hyperbolic 3-manifolds with special properties. Following the works presented in his doctoral thesis, the candidate aims to investigate the relation between the real and complex versions of the GLT conjectures (concerning uniformization in dimension four), the applications of the techniques of classic geometry applied to classical Teichmüller theory and its interplay with dynamical systems. (AU)