Mathematical Research Letters

Volume 25 (2018)

Number 5

A representation theoretic characterization of simple closed curves on a surface

Pages: 1485 – 1496



Thomas Koberda (Department of Mathematics, University of Virginia, Charlottesville, Va., U.S.A.)

Ramanujan Santharoubane (Department of Mathematics, University of Virginia, Charlottesville, Va., U.S.A.)


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 $\pi_1(S)$ has a representative in its free homotopy class which is 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.

The first author was partially supported by Simons Foundation Collaboration Grant number 429836, and is partially supported by NSF Grant DMS-1711488 and by an Alfred P. Sloan Foundation Research Fellowship.

Received 16 March 2016

Accepted 28 June 2017

Published 1 February 2019