Uniform Unicellular Dessins d'Enfants with Trivial Automorphism Groups

Source abstract

A Belyĭ function on a smooth projective algebraic curve defined over a number field determines a bipartite graph called a dessin d'enfant. We study the regularity and automorphism groups of dessins with uniform passports. In previous papers, we proved that every passport of the form $[n,n,n]$, $[n,b^{q},n]$, or $[b^{q},b^{q},n]$ of genus at least $2$ admits a dessin with trivial automorphism group. In this paper, we prove the analogous result for passports of the form $[a^{p},b^{q},n]$. The proof is mainly based on a counting argument: we compare a lower bound for the number of permutation representations having the prescribed passport with an upper bound for the number admitting a nontrivial automorphism. Together with our previous results, this shows that every uniform unicellular passport of genus at least $2$ admits a dessin with trivial automorphism group.