A counterexample to the simple loop conjecture for PSL(2,R)

TL;DR

This paper provides an explicit, elementary counterexample to the simple loop conjecture for surface group representations into PSL(2,R), showing the existence of many non-injective homomorphisms that do not kill simple closed curves.

Contribution

It constructs explicit, elementary examples of non-injective representations of surface groups into PSL(2,R) that do not kill simple closed curves, countering the simple loop conjecture.

Findings

Existence of uncountably many such homomorphisms

Homomorphisms are non-conjugate and non-injective

Homomorphisms do not kill simple closed curves

Abstract

In this note, we give an explicit counterexample to the simple loop conjecture for representations of surface groups into PSL(2,R). Specifically, we show that for any surface with negative Euler characteristic and genus at least 1, there are uncountably many non-conjugate, non-injective homomorphisms of its fundamental group into PSL(2,R) that kill no simple closed curve (nor any power of a simple closed curve). This result is not new -- work of Louder and Calegari for representations of surface groups into SL(2, C) applies to the PSL(2,R) case, but our approach here is explicit and elementary.