Moira Chas, Stony Brook University

Abstract:
This work is a consequence of the close connection between combinatorial group theory and the topology of surfaces. In the eighties Goldman discovered a Lie algebra structure on the vector space with given basis the set of conjugacy classes of the fundamental group of an oriented surface. Recall that the set of conjugacy classes of the fundamental group is in one-to-one correspondence with the set of free homotopy classes of oriented curves on that surface. The Lie bracket of two conjugacy classes a and b denoted [a,b] is a sum over the intersection points of representatives of a and of b. Here we assume the representatives are transversal, and we take for each intersection point the usual loop product based at that point with a sign given by the intersection number at that point. If one of the classes has a simple (that is, without self-intersection) representative we give a combinatorial group theory description of the terms of the Lie bracket in the given basis. Also using combinatorial group theory tools we show that certain pairs of elements of of the fundamental group of the surface are not conjugate. Combining this last statement, with the description of the Lie bracket, we obtain that the bracket has as many terms, counted with multiplicity, as the minimal number of intersection points of a and b. In other words, the bracket with a simple element determines minimal intersection numbers.