Simple Loops on Surfaces and Their Intersection Numbers

Given a compact orientable surface , let S(() be the set of isotopy classes of essential simple loops on. We determine a complete set of relations for a function from S(() to Z to be a geometric intersection number function. As a consequence, we obtain explicit equations in R S(() and P(R S(()) deening Thurston's space of measured laminations and Thurston's compactiication of the Teichm uller space. These equations are not only piecewise integral linear but also semi-real algebraic. Given a compact orientable surface == g;r of genus g with r boundary components , let S = S(() be the set of isotopy classes of essential simple loops on. A function f : S(() ! R is called a geometric intersection number function, or simply geometric function if there is a measured lamination m on so that f() is the measure of in m. Geometric functions were introduced and studied by W. Thurston in his work on the classiication of surface homeomorphisms and the compactiication of the Teichm uller spaces ((FLP], Th]). The space of all geometric functions under the pointwise convergence topology is homeomorphic to Thurston's space of measured laminations ML((). Thurston showed that ML(() is homeomorphic to a Euclidean space and ML(() has a piecewise integral linear structure invariant under the action of the mapping class group. The projectiviza-tion of ML(() is Thurston's boundary of the Teichm uller space. The object of the paper is to characterize all geometric functions on S((). As a consequence, both ML(() and its projectivization are reconstructed explicitly in terms of an intrinsic combinatorial structure on S(().