A representation theoretic characterization of simple closed curves on a surface

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.