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Belyi function whose Grothendieck dessin is a flower tree with two ramification indices. (English) Zbl 1096.14024
The present paper considers Grothendieck dessins in the plane looking like flower trees, i.e. with graph diameter \(4\), a vertex of valency \(r\) as midpoint whose \(k+l\) neighbour vertices have \(k\) times valency \(m\) and \(l\) times valency \(n\not=m\,\). The author gives a nice and explicit method to determine the Belyi function belonging to this dessin and proves that the moduli field of the dessin is a field of definition for the Belyi function. The result generalises earlier contributions by L. Schneps [in: The Grothendieck theory of dessins d’enfants. Lond. Math. Soc. Lect. Note Ser. 200, 47–77 (1994; Zbl 0823.14017)], G. Shabat and A. Zvonkin [Contemp. Math. 178, 233–275 (1994; Zbl 0816.05024)] and L. Zapponi [Compos. Math. 122, No. 2, 113–133 (2000; Zbl 0968.14011)].
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H30 Coverings of curves, fundamental group