A counterexample to the simple loop conjecture for $\mathrm{PSL}(2,\mathbb{R})$

Kathryn Mann

Vol. 269 (2014), No. 2, 425–432

DOI: 10.2140/pjm.2014.269.425

Abstract

In this note, we give an explicit counterexample to the simple loop conjecture for representations of surface groups into $PSL\left(2,ℝ\right)$. Specifically, we use a construction of DeBlois and Kent to show that for any orientable surface with negative Euler characteristic and genus at least 1, there are uncountably many nonconjugate, noninjective homomorphisms of its fundamental group into $PSL\left(2,ℝ\right)$ 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 $SL\left(2,ℂ\right)$ applies to the $PSL\left(2,ℝ\right)$ case, but our approach here is explicit and elementary.

Keywords

simple loop conjecture, surface group, character variety, representation, representation space, $ \mathrm{PSL}(2,\mathbb{R})$

Mathematical Subject Classification 2010

Primary: 57M05

Secondary: 57N16

Milestones

Received: 27 November 2012

Revised: 4 September 2013

Accepted: 9 September 2013

Published: 26 July 2014