Maximal Representations of Surface Groups: Symplectic Anosov Structures

TL;DR

This paper surveys maximal surface group representations into Hermitian Lie groups, focusing on symplectic cases, showing they produce discrete faithful Anosov representations with limit sets as rectifiable circles.

Contribution

It provides a detailed analysis of maximal representations into symplectic groups, highlighting their Anosov properties and geometric limit set structures.

Findings

Maximal representations are discrete faithful and realize surface groups as Kleinian groups.

The limit set of these representations is a rectifiable circle.

Maximality is characterized by the Toledo invariant.

Abstract

Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Gamma be the fundamental group of a compact orientable surface of genus at least 2. We survey the study of maximal representations, that is the subset of Hom(Gamma,G) which is a union of components characterized by the maximality of the Toledo invariant. Then we concentrate on the particular case G=SP(2n,R), and we show that the image of Gamma under any maximal representation is a discrete faithful realization of Gamma as a Kleinian group of complex motions in X with an associated Anosov system, and whose limit set in an appropriate compactification of X is a rectifiable circle.