[Paper Review] On the Torelli group action on compact character varieties

Yohann Bouilly|arXiv (Cornell University)|Jan 23, 2020

Advanced Algebra and Geometry10 references4 citations

TL;DR

This paper provides a new proof of the ergodicity of the Torelli group action on compact G-character varieties for connected, semi-simple, and compact Lie groups G, extending Funar and Marché's result for G=SU(2). Using Fox calculus and submersion arguments on the tangent space, it establishes that the Torelli group acts ergodically on each connected component of the SU(n)-character variety and on products of such varieties, generalizing previous results on mapping class group actions.

ABSTRACT

The aim of this article is to prove that the Torelli group action on the G-character varieties is ergodic for G a connected, semi-simple and compact Lie group.

Motivation & Objective

• To provide an alternative proof of the ergodicity of the Torelli group action on SU(2)-character varieties, previously established by Funar and Marché.
• To extend this ergodicity result to higher rank compact Lie groups G, specifically SU(n) for n ≥ 2.
• To generalize the result to products of character varieties, showing weak mixing of the Torelli group action.
• To establish that the Torelli group acts ergodically on each connected component of the G-character variety for connected, semi-simple, compact G.

Proposed method

• Utilizes Fox calculus to compute derivatives of group word maps in the Lie algebra, enabling explicit computation of tangent maps.
• Constructs one-parameter families of group elements (ρ_t(a₁)) to deform representations and analyze the action on the character variety.
• Analyzes the differential of a key map K: G⁴ × ℝ → G × G at a specific point to prove it is a submersion.
• Applies the fact that the rank of the differential equals the codimension of the centralizer, which is finite and discrete for regular representations.
• Uses the symplectic structure on the character variety defined via the Killing form and group cohomology H¹(Γ, g_ρ).
• Leverages the known isomorphism H²(Γ, ℝ) ≅ ℝ to define the symplectic form ω_G via the pairing ⟨u(γ₁), Ad(ρ(γ₁))v(γ₂)⟩.

Experimental results

Research questions

• RQ1 Does the Torelli group act ergodically on the SU(n)-character variety for n ≥ 2, extending the SU(2) case?
• RQ2 Can the ergodicity of the Torelli group action be generalized beyond SU(2) to other connected, semi-simple, compact Lie groups G?
• RQ3 Is the action of the Torelli group on products of G-character varieties weakly mixing?
• RQ4 What is the role of the centralizer in determining the rank of the differential of the deformation map K?
• RQ5 How can Fox calculus be used to establish submersion properties of maps on character varieties?

Key findings

• The Torelli group acts ergodically on the SU(n)-character variety M(Γ, SU(n)) with respect to the Goldman symplectic measure for all n ≥ 2.
• For any connected, semi-simple, compact Lie group G, the Torelli group acts ergodically on each connected component of M(Γ, G).
• The action of the Torelli group on the product space M(Γ, G)^k is weakly mixing for all k ≥ 1.
• The key map K is a submersion at the identity point, which implies the image is open and the action is ergodic.
• The differential of the deformation map has rank equal to the codimension of the centralizer, ensuring transversality in the tangent space.
• The proof relies on the discrete centralizer condition for regular points in M(Γ, G), which holds due to the semi-simplicity and compactness of G.