On normalizations of Thurston measure on the space of measured laminations (Q2329349)

The space \(\mathcal{ML}(\Sigma)\) of measured laminations of a surface \(\Sigma\) of genus \(g\) with \(n\) punctures is an integral piecewise linear manifold of dimension \(6g-6+2n\). \textit{H. Masur} showed [Proc. Am. Math. Soc. 94, 455--459 (1985; Zbl 0579.32035)] that there exists a measure on \(\mathcal{ML}(\Sigma)\), unique up to scaling, which is invariant under the action of the mapping class group. The authors consider two different ways to normalize this measure: the first normalization \(\mu_\omega\) (``the symplectic Thurston measure'') comes from the symplectic form on \(\mathcal{ML}(\Sigma)\) as defined by Thurston, the second normalization \(\mu_{\mathbb{Z}}\) (``the integer Thurston measure'') from the integral affine structure on \(\mathcal{ML}(\Sigma)\) (which is natural for the counting of hyperbolic geodesics and counting problems on the space of quadratic differentials). In the present paper, answering a question of \textit{K. Rafi} and \textit{J. Souto} [Geom. Funct. Anal. 29, No. 3, 871--889 (2019; Zbl 1420.30012)], the authors determine the ratio between these two measures (explicitly, \(\mu_\omega/\mu_\mathbb{Z} = 2^{2g+n-3}\)). As the authors note, this result was obtained independently, by different methods, also by \textit{F. Arana-Herrera} [``Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics'', Preprint, \url{arXiv:1902.05626}] and by \textit{V. Delecroix} et al., [``Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves'', Preprint, \url{arXiv:1908.08611}].