# On normalizations of Thurston measure on the space of measured laminations

## TL;DR

This paper investigates different normalizations of Thurston measure on the space of measured laminations, providing a precise ratio between two common measures used in the study of hyperbolic surfaces.

## Contribution

It explicitly computes the ratio between two natural normalizations of Thurston measure on measured lamination space, clarifying their relationship.

## Key findings

- Derived the ratio between two Thurston measure normalizations
- Provided explicit formulas for measure comparisons
- Enhanced understanding of measure normalization in Teichmüller theory

## Abstract

The space of measured laminations $\mathcal{ML}(\Sigma)$ associated to a topological surface $\Sigma$ of genus $g$ with $n$ punctures is an integral piecewise linear manifold of real dimension $6g-6+2n$. There is also a natural symplectic structure on $\mathcal{ML}(\Sigma)$ defined by Thurston. The integral and symplectic structures define a pair of measures on $\mathcal{ML}(\Sigma)$ which are known to be proportional. The projective class of these measures on $\mathcal{ML}(\Sigma)$ is called the Thurston measure. In this note we compute the ratio between two normailzations of the Thurston measure.