6 J an 1 99 8 Simple Loops on Surfaces and Their Intersection Numbers

Given a compact orientable surface Σ, let S(Σ) be the set of isotopy classes of essential simple loops on Σ. We determine a complete set of relations for a function from S(Σ) to Z to be a geometric intersection number function. As a consequence, we obtain explicit equations in R S(Σ) and P (R S(Σ)) defining Thurston's space of measured laminations and Thurston's compactification of the Teichmüller space. These equations are not only piecewise integral linear but also semi-real algebraic. Given a compact orientable surface Σ =Σ g,r of genus g with r boundary components , let S = S(Σ) be the set of isotopy classes of essential simple loops on Σ. A function f : S(Σ) → R is called a geometric intersection number function, or simply geometric function if there is a measured lamination m on Σ so that f (α) is the measure of α in m. Geometric functions were introduced and studied by W. Thurston in his work on the classification of surface homeomorphisms and the compactification of the Teichmüller spaces ([FLP], [Th]). The space of all geometric functions under the pointwise convergence topology is homeomorphic to Thurston's space of measured laminations ML(Σ). Thurston showed that ML(Σ) is homeomorphic to a Euclidean space and ML(Σ) has a piecewise integral linear structure invariant under the action of the mapping class group. The projectiviza-tion of ML(Σ) is Thurston's boundary of the Teichmüller space. The object of the paper is to characterize all geometric functions on S(Σ). As a consequence, both ML(Σ) and its projectivization are reconstructed explicitly in terms of an intrinsic (QP 1 , P SL(2, Z)) structure on S(Σ). 1 Theorem 1. Suppose Σ is a compact orientable surface of negative Euler number. Then a function f on S(Σ) is geometric if and only if for each incompressible subsurface Σ Recall that a subsurface Σ ′ ⊂ Σ is incompressible if each essential loop in Σ ′ is still essential in Σ. It is well known that if each boundary component of Σ ′ is essential in Σ, then Σ ′ is essential. Geometric functions and measures laminations haven been studied from many different points of views. Especially, they are identified with height functions and horizontal foliations associated to holomorphic quadratic forms on Σ ([Ga], [HM], [Ker1]). They are also related to the translation length functions of group action on R-trees and the valuation theory ([Bu], …