Mathematical Research Letters

Volume 25 (2018)

Number 5

A representation theoretic characterization of simple closed curves on a surface

Pages: 1485 – 1496

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n5.a6

Authors

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.)

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 $\pi_1(S)$ 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.

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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