Totally elliptic surface group representations

TL;DR

This paper characterizes totally elliptic surface group representations into 4 ext{R} ext{ and }4 ext{C} ext{, showing they are either compact or Deroin--Tholozan representations, thus classifying these special representations.

Contribution

It provides a complete classification of totally elliptic surface group representations into 4 ext{R} ext{ and }4 ext{C} ext{, identifying their precise structure and types.

Findings

All such representations are either into a compact subgroup or are Deroin--Tholozan representations.

The characterization applies specifically to representations into 4 ext{R} ext{ and }4 ext{C} ext{}.

The paper offers a complete description of totally elliptic surface group representations.

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.