Simple loops on surfaces and their intersection numbers

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 $\Bbb Z$ to be a geometric intersection number function. As a consequence, we obtain explicit equations in $\Bbb R^{\Cal S(\Sigma)}$ and $P (\Bbb R^{\Cal S(\Sigma)})$ 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.

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Published 1 January 1998