Simple loop conjecture for discrete representations in PSL$(2,\,\mathbb R)$

TL;DR

This paper proves that for any discrete but non-faithful representation of a surface group into PSL(2,R), there exists a simple closed curve in its kernel, confirming the Simple Loop Conjecture in this setting.

Contribution

It establishes the Simple Loop Conjecture for all discrete, non-faithful representations of surface groups into PSL(2,R).

Findings

Existence of simple closed curves in the kernel of such representations.

Confirmation of the Simple Loop Conjecture for these cases.

Abstract

We show that the Simple Loop Conjecture holds for any representation $\rho\colon\pi_1(S)\longrightarrow \text{PSL}(2,\,\mathbb R)$ that is discrete but not faithful. That is, we show the existence of a simple closed curve in the kernel of such a representation.