[Paper Review] Rigidity and geometricity for surface group actions on the circle

Kathryn Mann, Maxime Wolff|arXiv (Cornell University)|Oct 13, 2017

Geometric and Algebraic Topology3 references4 citations

TL;DR

This paper establishes that all rigid representations of surface groups into the group of orientation-preserving circle homeomorphisms are geometric, meaning they are semi-conjugate to discrete, faithful actions in PSL(2,ℝ). Using bending deformations and analysis of rotation numbers and bad tori, the authors prove that path-rigid representations—those with no nontrivial deformations—must arise from geometric structures, resolving a key question in circle dynamics and character varieties.

ABSTRACT

We prove that rigid representations of the fundamental group of a surface into the group of oreintation-preserving homeomorphisms of the circle are geometric, thereby establishing a converse statement of a theorem by the first author.

Motivation & Objective

• To establish a converse to a prior result by Mann (2015), showing that rigid representations of surface groups in Homeo⁺(S¹) are geometric.
• To resolve the question of whether path-rigid representations (those with no nontrivial deformations) must arise from geometric structures.
• To analyze the role of rotation numbers and 'bad tori' in obstructing or enabling deformations of surface group actions on the circle.
• To explore the topological structure of the character space X(Γg, Homeo⁺(S¹)) and its relation to classical character varieties.
• To investigate whether bending deformations generate all paths in the representation space, drawing analogies to Thurston's earthquake theorem.

Proposed method

• Use bending deformations along simple closed curves to construct continuous paths in the representation space Hom(Γg, Homeo⁺(S¹)).
• Analyze rotation numbers of elements in surface group representations, particularly for nonseparating curves, using results from Antonov on random word rotation numbers.
• Define and study 'bad tori'—one-holed torus subgroups where all nonseparating curves have rational rotation numbers—as obstructions to deformations.
• Apply the theory of semi-conjugacy and character spaces to relate topological rigidity to geometric structures via the largest Hausdorff quotient X(Γg, Homeo⁺(S¹)).
• Use the fact that representations with maximal Euler number (±(2g−2)) are rigid and semi-conjugate to PSL(2,ℝ) actions, as established by Matsumoto.
• Employ techniques from geometric invariant theory and GIT quotients to understand the structure of the character space as a Hausdorff quotient of Hom(Γg, Homeo⁺(S¹))/Homeo⁺(S¹).

Experimental results

Research questions

• RQ1 Are all rigid representations of surface groups in Homeo⁺(S¹) geometric, i.e., semi-conjugate to discrete, faithful actions in PSL(2,ℝ)?
• RQ2 Can every path in the representation space Hom(Γg, Homeo⁺(S¹)) be realized via bending deformations along simple closed curves?
• RQ3 Do there exist path-rigid representations containing at least one 'bad torus'—a one-holed torus subgroup where all nonseparating curves have rational rotation numbers?
• RQ4 Is there a representation of Γg in Homeo⁺(S¹) that is locally rigid but not rigid, indicating a distinction between local and global rigidity?
• RQ5 Does the Euler number classify connected components of X(Γg, Homeo⁺(S¹)), as it does in the PSL(2,ℝ) case?

Key findings

• All rigid representations of π₁Σg in Homeo⁺(S¹) are geometric, meaning they are semi-conjugate to discrete, faithful representations into PSL(2,ℝ).
• Path-rigid representations cannot contain two disjoint bad tori, and the absence of any bad torus would imply geometricity via an enhanced version of Lemma 5.11.
• Representations with maximal Euler number (±(2g−2)) are rigid and semi-conjugate to PSL(2,ℝ) actions, as shown by Matsumoto.
• The character space X(Γg, Homeo⁺(S¹)) is the largest Hausdorff quotient of Hom(Γg, Homeo⁺(S¹))/Homeo⁺(S¹), parameterizing semi-conjugacy classes of actions.
• Bending deformations along simple closed curves generate paths in the representation space, suggesting a structural analogy to Teichmüller theory and Thurston's earthquake theorem.
• There exist minimal actions of the free group on S¹ where almost all random words have rational rotation numbers, as per Antonov's theorem, but the existence of a non-abelian minimal action with all nonseparating curves having rational rotation numbers remains open.