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# Mathematical Research Letters

## Volume 5 (1998)

### Number 1

### Simple loops on surfaces and their intersection numbers

Pages: 47 – 56

DOI: http://dx.doi.org/10.4310/MRL.1998.v5.n1.a4

#### Author

#### 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 $\Bbb Z$ 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.