An algebraic characterization of simple closed curves on surfaces with boundary

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.