Minimal intersection of curves on surfaces

TL;DR

This paper provides a combinatorial description of Goldman’s Lie bracket on curves on surfaces, showing it directly encodes minimal intersection numbers and exploring applications in Teichmüller theory and mapping class groups.

Contribution

It offers a new combinatorial group theory approach to Goldman’s Lie bracket, proving it reflects minimal intersection points and extending results to unoriented curves.

Findings

Lie bracket terms match minimal intersection points

No cancellation occurs with simple elements

Applications to Teichmüller space and mapping class groups

Abstract

In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points of a and b of the loop product of at the intersection points. If one of the classes has a simple representative we give a combinatorial group theory description of the terms of the Lie bracket and prove that this 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 has no cancellation and determines minimal intersection numbers. We show that analogous results hold for the Lie bracket (also discovered by Goldman) of unoriented curves. We give three applications: a factorization of Thurston's map defining the boundary of Teichmuller space, various decompositions of the underlying vector space of conjugacy classes into ad invariant subspaces and a connection between bijections of the set of conjugacy classes of curves on a surface preserving the Goldman bracket and the mapping class group.