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Gassmann equivalent dessins. (English) Zbl 1246.11126
Summary: This article studies pairs of dessins d’enfants that arise from Gassmann triples of groups (\(G, H, H^{\prime}\)) together with pairs \((g_{0}, g_{1})\) of elements in \(G\). We show that the two dessins have isomorphic monodromy groups, have the same branching data and the same number of components. Moreover, the sums of the genera of the components of the two dessins are the same. We give an example where the individual genera of the components of the first dessin differ from the genera of the components of the second dessin.

MSC:
11G32 Arithmetic aspects of dessins d’enfants, Belyĭ theory
14H57 Dessins d’enfants theory
20B05 General theory for finite permutation groups
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[1] Gassmann F., Math. Z. 25 pp 124– (1926)
[2] DOI: 10.1016/0022-314X(77)90070-1 · Zbl 0389.12006 · doi:10.1016/0022-314X(77)90070-1
[3] DOI: 10.1016/0022-314X(78)90020-3 · Zbl 0393.12009 · doi:10.1016/0022-314X(78)90020-3
[4] Schneps L., The Grothendieck Theory of Dessins d’Enfants (1994) · Zbl 0798.00001
[5] Grothendieck , A. ( 1984 ).Esquisse d’un Programme.In ”Geometric Galois Actions” (L. Schneps, P. Lochak, eds.) Vol. 1 ”Around Grothendieck’s Esguisse d’un Programme.” London Math Soc. Lecture Notes Series. Cambridge Univ. Press, 1997, Vol. 242, 5–48 (English version, same volume, 243–284) .
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