[Paper Review] Mapping class group dynamics on Aff(C)-characters

Selim Ghazouani|arXiv (Cornell University)|Jun 29, 2014

Mathematical Dynamics and Fractals8 references3 citations

TL;DR

This paper establishes that the mapping class group acts ergodically on the character variety of representations from the fundamental group of a genus $ g \geq 2 $ surface into the complex affine group $ \mathrm{Aff}(\mathbb{C}) $. By analyzing the action on the linear part $ \mathrm{H}^1(S, \mathbb{C}^*) \simeq (\mathbb{C}^*)^{2g} $ and the Torelli group's action on twisted cohomology fibers $ \mathrm{PH}^1_\alpha(\Gamma, \mathbb{C}) $, the authors prove ergodicity, implying that almost every such representation arises as the holonomy of a branched affine structure on the surface.

ABSTRACT

We prove that in genus bigger than $2$, the mapping class group action on $\mathrm{Aff}(\mathbb{C})$-characters is ergodic. This implies that almost every representation $π_1 S \longrightarrow \mathrm{Aff}(\mathbb{C})$ is the holonomy of a branched affine structure on $S$, where $S$ is a closed orientable surface of genus $g \geq 2$.

Motivation & Objective

• To establish ergodicity of the mapping class group action on the $ \mathrm{Aff}(\mathbb{C}) $-character variety for surfaces of genus $ g \geq 2 $.
• To understand the dynamics of the mapping class group on representations $ \pi_1S \to \mathrm{Aff}(\mathbb{C}) $, particularly via the Torelli group's action on twisted cohomology fibers.
• To resolve the holonomy problem for branched complex affine structures by showing that the set of geometric holonomies is of full measure in the character variety.

Proposed method

• The character variety is structured as a fiber bundle over $ (\mathbb{C}^*)^{2g} $, with fibers isomorphic to $ \mathbb{CP}^{2g-3} $, parametrized by twisted cohomology $ \mathrm{H}^1_\alpha(\Gamma, \mathbb{C}) $.
• Ergodicity of the $ \mathrm{Sp}(2g, \mathbb{Z}) $-action on $ (\mathbb{C}^*)^{2g} $ is established via Moore’s theorem on diagonal actions on $ \mathbb{R}^{2g} \times (\mathbb{R}/\mathbb{Z})^{2g} $.
• Explicit computation of Dehn twist actions on $ \mathrm{PH}^1_\alpha(\Gamma, \mathbb{C}) $ shows that the Torelli group acts ergodically on the fibers for almost all $ \alpha \in (\mathbb{C}^*)^{2g} $.
• The proof combines ergodicity on the base with fiberwise ergodicity of the Torelli group, leveraging projective representations $ \tau_\alpha: \mathcal{I}(S) \to \mathrm{PGL}(2g-2, \mathbb{C}) $.
• The action preserves no absolutely continuous measure, contrasting with reductive cases where Goldman’s symplectic form ensures invariant measures.

Experimental results

Research questions

• RQ1 Which representations $ \pi_1S \to \mathrm{Aff}(\mathbb{C}) $ arise as holonomies of branched affine structures?
• RQ2 Does the mapping class group action on the $ \mathrm{Aff}(\mathbb{C}) $-character variety admit a Ratner-type structure or non-homogeneous orbit closures?
• RQ3 For which $ \alpha \in (\mathbb{C}^*)^{2g} $ is the Torelli group representation $ \tau_\alpha: \mathcal{I}(S) \to \mathrm{PGL}(2g-2, \mathbb{C}) $ discrete?
• RQ4 Can the Torelli group representations $ \tau_\alpha $ yield lattices in $ \mathrm{PU}(g-1,g-1) $ when $ \alpha $ is unitary?
• RQ5 How do directional foliations behave in branched affine structures with real-linear holonomy?

Key findings

• The mapping class group action on the $ \mathrm{Aff}(\mathbb{C}) $-character variety is ergodic for all $ g \geq 2 $, a result proven by combining ergodicity on the base $ (\mathbb{C}^*)^{2g} $ and fiberwise ergodicity of the Torelli group.
• The Torelli group acts ergodically on the fibers $ \mathrm{PH}^1_\alpha(\Gamma, \mathbb{C}) \simeq \mathbb{CP}^{2g-3} $ for almost every $ \alpha \in (\mathbb{C}^*)^{2g} $, as shown via explicit Dehn twist computations.
• The action preserves no absolutely continuous measure relative to Lebesgue measure, distinguishing it from reductive character varieties where such measures exist.
• The set of representations that arise as holonomies of branched affine structures is an open subset of full measure in the character variety.
• The representation $ \tau_\alpha: \mathcal{I}(S) \to \mathrm{PGL}(2g-2, \mathbb{C}) $ is a projective representation arising from the Torelli group's action on twisted cohomology.
• A conjecture is proposed: every strictly affine representation (non-unitary, with angles generating an infinite subgroup of $ \mathbb{R}/\mathbb{Z} $) is realizable as a holonomy of a branched affine structure.