A representation theoretic characterization of simple closed curves on a surface

TL;DR

This paper constructs a sequence of finite-dimensional representations of a surface's fundamental group that distinguishes simple closed curves from non-simple ones, enabling an algorithm to identify simple closed curves via representation theory.

Contribution

It introduces a novel representation theoretic approach that combines TQFT ideas and LERF techniques to decide if a loop on a surface is homotopic to a simple closed curve.

Findings

All simple closed curves act with finite order in the constructed representations.

Non-simple closed curves act with infinite order in the limit.

Provides an algorithm to detect simple closed curves using representation theory.

Abstract

We produce a sequence of finite dimensional representations of the fundamental group $\pi_1(S)$ of a closed surface where all simple closed curves act with finite order, but where each non--simple closed curve eventually acts with infinite order. As a consequence, we obtain a representation theoretic algorithm which decides whether or not a given element of $\pi_1(S)$ is freely homotopic to a simple closed curve. The construction of these representations combines ideas from TQFT representations of mapping class groups with effective versions of LERF for surface groups.