Combinatorics of simple closed curves on the twice punctured torus

Abstract

In the standard enumeration of homotopy classes of curves on a surface as words in a generating set for the fundamental group it is a very hard problem to discern those that are simple. In this paper we describe how the complex of simple closed curves on a twice punctured torus Σ may be given a strikingly simple description by representing them as homotopy classes ofpaths in agroupoid with two base points. Our starting point are the π₁-train tracks developed by Birman and Series. These are weighted train tracks parameterizing the simple closed curves on Σ similar to Thurston’s, but they are defined relative to a fixed presentation of π₁(Σ). We approach the problem by cutting the surface into two disjoint “cylinders”; this decomposes the π₁-train tracks into two disjoint parts, relative to which all patterns and relations become much more transparent, each part reducing essentially to the well-known case of a once punctured torus. We obtain global coordinates, called π_1,2-weights, for simple closed loops. These coordinates can be easily identified with Thurston’s projective measured lamination spaceS ³. We also solve the problem which originally motivated this work by proving a simple relationship between the leading terms of traces of simple loops in a holomorphic family of representationsρ: π₁(Σ) → PSL(2,C) (corresponding to the Maskit embedding of the twice punctured torus) and the π_1,2-weights.