Uniform Unicellular Dessins d'Enfants with Trivial Automorphism Groups

@misc{ohnishi2026uniform,
  title = {{Uniform Unicellular Dessins d'Enfants with Trivial Automorphism Groups}},
  author = {Tatsuya Ohnishi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33602},
  eprint = {2609.33602},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33602},
}

TY  - UNPB
TI  - Uniform Unicellular Dessins d'Enfants with Trivial Automorphism Groups
AU  - Tatsuya Ohnishi
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33602
UR  - https://arxiv.org/abs/2609.33602
AB  - A Bely\uı 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.
ER  -