Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bending, entropy and proper affine actions of surface groups

Martin Bridgeman, Richard Canary, Andres Sambarino

Abstract

We show that for any closed surface $S$ there is an explict neighborhood $V$ of the fuchsian locus in quasifuchsian space $\mathsf{QF}(S)$ such that for every representation $ρ\in V$ which is not fuchsian, there is a proper affine action on $\mathfrak{sl}(2,\mathbb{C})$ with linear part $\mathsf{Ad}(ρ)$. We further show that there is a larger neighborhood $U$ of the Fuchsian locus so that every critical point of the entropy function in $U$ lies on the Fuchsian locus.

Bending, entropy and proper affine actions of surface groups

Abstract

We show that for any closed surface there is an explict neighborhood of the fuchsian locus in quasifuchsian space such that for every representation which is not fuchsian, there is a proper affine action on with linear part . We further show that there is a larger neighborhood of the Fuchsian locus so that every critical point of the entropy function in lies on the Fuchsian locus.
Paper Structure (25 sections, 34 theorems, 194 equations, 4 figures)

This paper contains 25 sections, 34 theorems, 194 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Geodesic currents and measured laminations
  4. The convex core and bending laminations
  5. Complex length
  6. Bending deformations
  7. Variation of complex length
  8. Critical points of the entropy function
  9. Binding pairs of laminations and $\delta$-intersection number
  10. Proper affine actions with quasifuchsian holonomy
  11. The Margulis invariant
  12. Totally Anosov representations in $\mathsf{PSL}(d,\mathbb C)$
  13. A criterion for proper affine actions
  14. Proper affine actions on $\sl(2,\mathbb C)$
  15. Proper affine actions on other complex Lie groups
  16. ...and 10 more sections

Key Result

Theorem 1.1

Suppose that $[\rho]\in \operatorname{QF}(S)$ is not fuchsian and either $\Omega_+(\rho)$ or $\Omega_-(\rho)$ is moderately bent, then $[\rho]$ is not a critical point of the entropy function $h$.

Figures (4)

  • Figure 1: Domain $\Omega_\nu$
  • Figure 2: Plane $L$
  • Figure 3: Functions $a_L(\theta)$ and $r_L(\theta)$ for $L \leq 1$
  • Figure 4: Plot of $r_1(\theta)$ on $[0,\frac{\pi}{2}]$

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • ...and 49 more