TITLE: Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants?
QUESTION [15 upvotes]: 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.

REPLY [5 votes]: If I am not mistaken, this precise issue was attacked by Dirk Smit during 1990s in a series of papers on Communications of Mathematical Physics. I am not sure why his papers are not quoted more widely in the number theory community nowadays. One reason may be he joined industry. But experts in my social circle are certainly familiar with his work.
Some relevant links are His paper on height function over moduli spaces using triangulations and His paper on obtaining "critical" string measure (for the dimension=26 case?) as well as an earlier paper on a continuous version of the height function (see the end).
I would suggest to start reading from the third paper and moving back to the first and second paper. The first paper basically answers your question and gives much more (not just how to triangulate the surface and how this is related to dessins d'enfants, but also the distribution of the height function near the boundary of $\mathcal{M}_g$). A general survey by Yuri Manin can be found via this link.
I should add some suggestions on questions like "It has been about 30 years...So what next???". To follow up on Dirk's work, I would suggest to focus on explicit examples like $g=2,3$ and do some concrete calculations. Recently a paper independently replicating a lot of Dirk's result (https://arxiv.org/abs/1902.02420) was  published. I do not think the discrepancy he raised in the end of the first paper was ever addressed, though.