Group actions and scattering problems in Teichm\"uller theory

TL;DR

This paper explores the intersection of Teichmüller theory, surface group actions, and Lorentzian symmetric spaces, offering a new perspective that simplifies understanding of moduli spaces of Riemann surfaces.

Contribution

It introduces a framework connecting surface group actions in Lorentzian symmetric spaces to Teichmüller theory, clarifying known results and highlighting open problems.

Findings

Simplified proofs of classical Teichmüller results

New insights into surface group actions in Lorentzian spaces

Organized framework for future research in the area

Abstract

In recent years, Teichm\"uller theory, which is the study of moduli spaces of marked Riemann surfaces, has come to be considered more and more from the point of view of actions of surface groups inside certain semi-simple Lie groups. In particular, we consider the case where the Lie groups in question have symmetric spaces which are lorentzian spacetimes. Indeed, this can be considered as the starting point of Mess' seminal work, which led to the development of new and strikingly simpler proofs of many results of Teichm\"uller theory by considering them in terms of geometric objects inside these symmetric spaces. Our aim is to provide a brief and straightforward introduction to this approach, whilst developing what we consider to be a useful mental framework for organising known results and open problems.