Dehn-Thurston Coordinates for Curves on Surfaces

Abstract

1.1. We study the space of isotopy classes of 1-dimensional submanifolds in a surface in this paper. The subject was originated by Max Dehn in his 1938 paper [De]. In this work, Dehn laid the foundation for the studies of the mapping class group of a surface and the space of isotopy classes of 1-dimensional submanifolds in a surface. According to Dehn, the most important proper 1-dimensional submanifolds are the curve systems which have the property that no component of the submanifold is null homotopic or homotopic into the boundary of the surface by homotopies relative to the boundary. Dehn defined the arithmetic field of a surface Σ, denoted by CS(Σ), to be the set of all isotopy classes of curve systems. Dehn’s main focus of study in [De] was the action of the mapping class group on the ”arithmetic field” CS(Σ). In 1976, William Thurston independently rediscovered the space CS(Σ) and put one more vital ingredient into the study of ”arithmetic field” CS(Σ). Namely, the geometric intersection numbers between two isotopy classes of 1-dimensional submanifolds. Recall that if α and β are isotopy classes of proper 1-dimensional submanifolds, then their geometric intersection number, denoted by I(α, β), is the minimal number of intersections between their representatives, i.e.,