Introduction

Let Σ be a (possibly punctured) surface of negative Euler characteristic, and let C(Σ) be the set of isotopy classes of families of disjoint simple closed curves on Σ. When Σ is a once punctured torus Σ1, there is a well known recursive structure on C(Σ1) which arises from the relationships between C(Σ1) (identified with the extended rational numbers), continued fractions, and PSL(2, ℤ) (the mapping class group of Σ1). The results in this paper arose out of a search for an analogous structure on C(Σ2), where Σ2 is a torus with two punctures. Masur and Minsky have recently described an alternative approach. Our method is motivated by the Bowen-Series construction of Markov maps for Fuchsian groups. This generalised the relationship between PSL(2, ℤ) (now thought of as a Fuchsian group acting in the hyperbolic plane) and continued fractions (now thought of as points in the limit set of PSL(2, ℤ)), to a large class of Fuchsian groups Γ. The Markov map was a map on the boundary at infinity, in other words the limit set Λ(Γ), which generated continued fraction expansions for points in Λ(Γ), and whose admissible sequences simultaneously gave an elegant solution to the word problem in Γ (see Section 1.1 below).

The idea behind this paper rests on the analogy between Γ acting on the hyperbolic plane and the mapping class group acting on Teichmüller space.