How many quasiplatonic curves?

TL;DR

This paper investigates the growth rate of the number of quasiplatonic Riemann surfaces and regular dessins, establishing that both grow approximately as g^{log g} for genus up to g.

Contribution

It provides the first asymptotic estimate for the number of quasiplatonic surfaces and regular dessins of bounded genus, revealing their growth behavior.

Findings

Number of quasiplatonic Riemann surfaces grows as g^{log g}

Number of regular dessins of genus ≤ g grows as g^{log g}

Establishes asymptotic growth rate for these classes

Abstract

We show that the number of isomorphism classes of quasiplatonic Riemann surfaces of genus $\,\le g\,$ has a growth of type $\,g^{\log g}\,$. The number of non--isomorphic regular dessins of genus $\,\le g\,$ has the same growth type.