An algebraic characterization of simple closed curves on surfaces with boundary

TL;DR

This paper provides an algebraic characterization of simple closed curves on surfaces with boundary using the Goldman Lie algebra, linking geometric properties to algebraic conditions and counting minimal self-intersections.

Contribution

It introduces a new algebraic criterion involving the Goldman Lie bracket to identify simple closed curves on surfaces with boundary, connecting combinatorial group theory with surface topology.

Findings

A conjugacy class contains an embedded representative iff its Goldman bracket with its cube is zero.

The paper counts the minimal self-intersection points of representatives of conjugacy classes.

Results relate algebraic properties to surface diffeomorphisms and classical conjectures.

Abstract

We characterize in terms of the Goldman Lie algebra which conjugacy classes in the fundamental group of a surface with non empty boundary are represented by simple closed curves. We prove the following: A non power conjugacy class X contains an embedded representative if and only if the Goldman Lie bracket of X with the third power of X is zero. The proof uses combinatorial group theory and Chas' combinatorial description of the bracket recast here in terms of an exposition of the Cohen-Lustig algorithm. Using results of Ivanov, Korkmaz and Luo there are corollaries characterizing which permutations of conjugacy classes are related to diffeomorphisms of the surfaces. The problem is motivated by a group theoretical statement from the sixties equivalent to the Poincare conjecture due to Jaco and Stallings and by a question of Turaev from the eighties. Our main theorem actually counts the minimal possible number of self-intersection points of representatives of a conjugacy class X in terms of the bracket of X with the third power of X.