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Rigidity and geometricity for surface group actions on the circle

Kathryn Mann, Maxime Wolff

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.

Rigidity and geometricity for surface group actions on the circle

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.

Paper Structure

This paper contains 32 sections, 66 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Character spaces and rigidity
  3. Geometric representations
  4. Results
  5. Strategy of the proof and organization of the article
  6. Preliminaries
  7. Based curves on surfaces
  8. Actions on the circle
  9. Basic dynamics of circle homeomorphisms
  10. One-parameter families and bending deformations
  11. Periodic sets under deformations
  12. The character space for $\mathbf \mathrm{Homeo}^+(S^1)$
  13. The Euler class
  14. A first statement
  15. Outline of Proof of Theorem \ref{['thm:SkImpliqueGeom']}
  16. ...and 17 more sections

Key Result

Theorem 1.4

In the space $\mathrm{Hom}(\Gamma_g,\mathrm{Homeo}^+(S^1))$, all geometric representations are rigid.

Figures (4)

  • Figure 1: Standard generators on the genus $g$ surface ($g=4$)
  • Figure 2: A directed chain of length $5$
  • Figure 3: A pair of pants with standard generators of its fundamental group
  • Figure 4: A decomposition of $\pi_1\Sigma_4$ into a graph of groups

Theorems & Definitions (137)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: KatieSurvey
  • Theorem 1.4: Mann KatieInvent
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.8
  • Proposition 1.9
  • Theorem 1.10
  • Proposition 1.11
  • ...and 127 more