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On Character Variety of Anosov Representations

Krishnendu Gongopadhyay, Tathagata Nayak

TL;DR

This work analyzes character varieties of Anosov representations of surface groups into ${ m SL}(n,oldsymbol{ olinebreak f C})$. It proves that the moduli spaces of irreducible $(Q^+,Q^-)$-Anosov and of Zariski-dense $(P^+,P^-)$-Anosov representations are complex manifolds with dimension $(2g+k-2)(n^2-1)$ for a $k$-punctured genus $g$ surface, and, in the closed-surface case, carry a natural holomorphic symplectic form. The approach combines Guichard–Wienhard Anosov theory with representation-variety geometry à la Sikora, showing open-ness and smoothness of the Anosov loci and establishing explicit dimension counts. For Γ = π1(Σ_g), the results yield holomorphic symplectic structures on the corresponding Anosov character varieties, providing a higher Teichmüller-type framework within complex geometry.

Abstract

Let $Γ$ 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 $Γ$ into $\SL(n , \C)$, $n \geq 2$, is a complex manifold of complex dimension \hbox{$(2g+k-2)(n^2-1)$}. For $Γ=π_1(Σ_g)$, we also show that these character varieties are holomorphic symplectic manifolds.

On Character Variety of Anosov Representations

TL;DR

This work analyzes character varieties of Anosov representations of surface groups into . It proves that the moduli spaces of irreducible -Anosov and of Zariski-dense -Anosov representations are complex manifolds with dimension for a -punctured genus surface, and, in the closed-surface case, carry a natural holomorphic symplectic form. The approach combines Guichard–Wienhard Anosov theory with representation-variety geometry à la Sikora, showing open-ness and smoothness of the Anosov loci and establishing explicit dimension counts. For Γ = π1(Σ_g), the results yield holomorphic symplectic structures on the corresponding Anosov character varieties, providing a higher Teichmüller-type framework within complex geometry.

Abstract

Let be the fundamental group of a -punctured, , closed connected orientable surface of genus . We show that the character variety of the -Anosov irreducible representations, resp. the character variety of the -Anosov Zariski dense representations of into , , is a complex manifold of complex dimension \hbox{}. For , we also show that these character varieties are holomorphic symplectic manifolds.
Paper Structure (8 sections, 25 theorems, 15 equations)

This paper contains 8 sections, 25 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Anosov Representations
  4. Varieties of Representations
  5. Character Varieties of Anosov representations
  6. Irreducible and Zariski dense representations
  7. Character Variety
  8. Symplecticity of Character Varieties of Anosov Representations

Key Result

Theorem 1.1

Let $\Gamma=F_k$, resp. $\pi_1(\Sigma_g)$, $k, g \geq 2$. The character variety of the $(Q^+, Q^-)$-Anosov irreducible representations, as well as the character variety of the $(P^+, P^-)$-Anosov Zariski dense representations of $\Gamma$ into ${\rm SL}(n , \mathbb C)$, $n \geq 2$, are complex manifo

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Example 2.7
  • Proposition 2.8
  • ...and 20 more