On Character Variety of Anosov Representations

Abstract

Let $\Gamma$ be the fundamental group of a $k$-punctured, $k\ge 0$, closed connected orientable surface of genus $g\ge 2$. 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 $\SL(n , \C)$, $n\ge 2$, 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.

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• Higgs bundles and higher Teichmüller spaces