Pairs of tree dessins, their Shabat polynomials, and monodromy groups

Benjamin Dupont; Revekka Kyriakoglou; Vassilis Metaftsis; Efstratios Prassidis; Alexandros Singh

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arXiv · 2510.10192

Pairs of tree dessins, their Shabat polynomials, and monodromy groups

Benjamin Dupont·Revekka Kyriakoglou·Vassilis Metaftsis·Efstratios Prassidis·Alexandros Singh

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Abstract

Coverings of the Riemann sphere by itself, ramified over two points, are given by so-called Shabat polynomials. The correspondence between Grothendieck's dessins d'enfants and Belyi maps then implies a bijection between Shabat polynomials and tree dessins (bicolored plane trees). Dessins can be assigned a combinatorial invariant known as their passport, which records the degrees of their vertices. We consider all possible passports determining a pair of tree dessins, determining the associated Shabat polynomials and monodromy groups.

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Benjamin Dupont, Revekka Kyriakoglou, Vassilis Metaftsis, Efstratios Prassidis, Alexandros Singh. 2025-10-11. Pairs of tree dessins, their Shabat polynomials, and monodromy groups. https://arxiv.org/abs/2510.10192

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