Higher Teichm\"uller Spaces: from SL(2,R) to other Lie groups

TL;DR

This paper surveys higher Teichm"uller spaces, extending classical concepts from PSL(2,R) to other Lie groups, emphasizing cohomological invariants and the structure of these spaces.

Contribution

It introduces a unified perspective on higher Teichm"uller spaces, highlighting cohomological invariants and structural properties for various Lie groups.

Findings

Characterizations of Teichm"uller space via cohomological invariants

Extension of classical invariants to Hermitian Lie groups

Structural insights into maximal and Hitchin representations

Abstract

The first part of this paper surveys several characterizations of Teichm\"uller space as a subset of the space of representation of the fundamental group of a surface into PSL(2,R). Special emphasis is put on (bounded) cohomological invariants which generalize when PSL(2,R) is replaced by a Lie group of Hermitian type. The second part discusses underlying structures of the two families of higher Teichm\"uller spaces, namely the space of maximal representations for Lie groups of Hermitian type and the space of Hitchin representations or positive representations for split real simple Lie groups.