Simple Loops on Surfaces and Their Intersection Numbers

Abstract

Given a compact orientable surface $\Sigma$, let $\Cal S(\Sigma)$ be the set of isotopy classes of essential simple loops on $\Sigma$. We determine a complete set of relations for a function from $\Cal S(\Sigma)$ to $\bold Z$ to be a geometric intersection number function. As a consequence, we obtain explicit equations in $\bold R^{\Cal S(\Sigma)}$ and $P(\bold R^{\Cal S(\Sigma)})$ defining Thurston's space of measured laminations and Thurston's compactification of the Teichm\"uller space. These equations are not only piecewise integral linear but also semi-real algebraic.