Belyi's theorem establishes a correspondence between smooth projective curves defined over number fields and the so called *dessins d'enfants* which are bipartite graphs embedded on an oriented surface so that each component of their complement is a topological disk. To be more precise, given a smooth projective curve $X$ of genus $g$ over $\overline{\Bbb{Q}}$, Belyi's theorem guarantees the existence of a branched cover $f:X(\Bbb{C})\rightarrow\Bbb{P}^1(\Bbb{C})$ unramified outside $\{0,1,\infty\}$. Then, $f^{-1}([0,1])$ is an embedded graph that can be considered as the $1$-skeleton of a CW structure on $X(\Bbb{C})$. Conversely, such a graph on an oriented compact surface of genus $g$ equips it with a complex structure and such a branched cover.

**My Question:** Can dessins be used to defined a "reasonable" (e.g. comparable to a Weil height) height function on the moduli space $\mathcal{M}\_g(\overline{\Bbb{Q}})$? For instance, is the minimum possible number of edges of a dessin (i.e. the minimum possible degree of a Belyi map on the complex curve) be used as a height function? Any reference to the literature on this question is highly appreciated.