Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Totally elliptic surface group representations

Arnaud Maret

TL;DR

The paper classifies totally elliptic surface group representations into $\mathrm{PSL}_2\mathbb{R}$ and $\mathrm{PSL}_2\mathbb{C}$, showing a dichotomy between compact (orthogonal) images and Deroin–Tholozan (DT) representations. For $\mathrm{PSL}_2\mathbb{R}$ with genus $g\ge1$ the only totally elliptic reduced representations are orthogonal; in genus zero they are DT unless the peripheral angle sum hits special values, yielding orthogonal cases. In $\mathrm{PSL}_2\mathbb{C}$, irreducibles are either unitary or DT-composed with $\mathrm{PSL}_2\mathbb{R}$, while reducibles are diagonal in $\mathrm{PSU}(2)$ except for certain punctured-sphere cases where reducible but non-unitary totally elliptic representations exist. The results connect to DT components, peripheral angles, Toledo numbers, and broader questions in the Bowditch–Goldman program, and extend the analysis to other Hermitian and non-Hermitian Lie groups. Overall, the work provides a precise, component-wise dichotomy for totally elliptic representations, clarifying when DT phenomena arise and how these representations sit inside relative character varieties with rich geometric and dynamical structure.

Abstract

A surface group representation into a Lie group is called totally elliptic if every simple closed curve on the surface is mapped to an elliptic element of the target group. In this note, we characterize all totally elliptic surface group representations into $\mathrm{PSL}_2\mathbb{R}$ and $\mathrm{PSL}_2\mathbb{C}$ by showing that they are either representations into a compact subgroup or Deroin--Tholozan representations.

Totally elliptic surface group representations

TL;DR

The paper classifies totally elliptic surface group representations into and , showing a dichotomy between compact (orthogonal) images and Deroin–Tholozan (DT) representations. For with genus the only totally elliptic reduced representations are orthogonal; in genus zero they are DT unless the peripheral angle sum hits special values, yielding orthogonal cases. In , irreducibles are either unitary or DT-composed with , while reducibles are diagonal in except for certain punctured-sphere cases where reducible but non-unitary totally elliptic representations exist. The results connect to DT components, peripheral angles, Toledo numbers, and broader questions in the Bowditch–Goldman program, and extend the analysis to other Hermitian and non-Hermitian Lie groups. Overall, the work provides a precise, component-wise dichotomy for totally elliptic representations, clarifying when DT phenomena arise and how these representations sit inside relative character varieties with rich geometric and dynamical structure.

Abstract

A surface group representation into a Lie group is called totally elliptic if every simple closed curve on the surface is mapped to an elliptic element of the target group. In this note, we characterize all totally elliptic surface group representations into and by showing that they are either representations into a compact subgroup or Deroin--Tholozan representations.

Paper Structure

This paper contains 17 sections, 22 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications and generalizations
  4. Motivations
  5. Perspectives away from $\operatorname{PSL}_2\mathop{\mathrm{\mathbb{C}}}\nolimits$
  6. Acknowledgements
  7. Totally elliptic representations in $\operatorname{PSL}_2 \mathop{\mathrm{\mathbb{R}}}\nolimits$
  8. Overview
  9. Positive genus
  10. Genus zero
  11. The case $n=3$
  12. The case $n=4$
  13. The case $n\geq 5$
  14. Totally elliptic representations in $\operatorname{PSL}_2\mathop{\mathrm{\mathbb{C}}}\nolimits$
  15. Overview
  16. ...and 2 more sections

Key Result

Theorem 1

Consider a surface $\Sigma$ of genus $g \geq 0$ with $n \geq 0$ punctures, and let $\rho\colon \pi_1\Sigma \to \operatorname{PSL}_2 \mathop{\mathrm{\mathbb{R}}}\nolimits$ be a reduced totally elliptic representation. If $\rho$ is not orthogonal, then $\Sigma$ must be a sphere with at least three pun

Figures (1)

  • Figure 1: A system of geometric generators for a surface $\Sigma$ of genus $g$ with $n$ punctures.

Theorems & Definitions (39)

  • Definition 1.1
  • Theorem 1
  • Corollary 2
  • Theorem 3: Propositions \ref{['prop-irreducible']} & \ref{['prop-reducible']}
  • Theorem 4: Example \ref{['example-reducible']}
  • Lemma 2.1: see e.g. beardon
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 29 more