2008/01/25 by Moira Chas, Chas, Moira, Fabiana Krongold +1 · 1 citation
Mathematics · #17B65 #57M99 #Advanced Combinatorial Mathematics #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.0801.3944
openalex publication_date 2008/01/25 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We characterize in terms of the Goldman Lie algebra which conjugacy classes\nin the fundamental group of a surface with non empty boundary are represented\nby simple closed curves. We prove the following: A non power conjugacy class X\ncontains an embedded representative if and only if the Goldman Lie bracket of X\nwith the third power of X is zero. The proof uses combinatorial group theory\nand Chas' combinatorial description of the bracket recast here in terms of an\nexposition of the Cohen-Lustig algorithm. Using results of Ivanov, Korkmaz and\nLuo there are corollaries characterizing which permutations of conjugacy\nclasses are related to diffeomorphisms of the surfaces. The problem is\nmotivated by a group theoretical statement from the sixties equivalent to the\nPoincare conjecture due to Jaco and Stallings and by a question of Turaev from\nthe eighties. Our main theorem actually counts the minimal possible number of\nself-intersection points of representatives of a conjugacy class X in terms of\nthe bracket of X with the third power of X.\n