On Character Variety of Anosov Representations

TL;DR

This paper investigates the structure of character varieties of Anosov representations of surface groups into SL(n,C), establishing their manifold and symplectic properties and computing their dimensions.

Contribution

It proves that these character varieties are complex manifolds with specific dimensions and are holomorphic symplectic for surface groups.

Findings

Character varieties are complex manifolds of dimension (2g+k-2)(n^2-1).

They are holomorphic symplectic manifolds for surface groups.

Results apply to both irreducible and Zariski dense Anosov representations.

Abstract

Let $\Gamma$ be the fundamental group of a $k$-punctured, $k \geq 0$, closed connected orientable surface of genus $g \geq 2$. We show that the character variety of the $(Q^+, Q^-)$-Anosov irreducible representations, resp. the character variety of the $(P^+, P^-)$-Anosov Zariski dense representations of $\Gamma$ into $\SL(n , \C)$, $n \geq 2$, is a complex manifold of complex dimension \hbox{$(2g+k-2)(n^2-1)$}. For $\Gamma=\pi_1(\Sigma_g)$, we also show that these character varieties are holomorphic symplectic manifolds.