Complete regular dessins and skew-morphisms of cyclic groups

Yan-Quan Feng, Kan Hu, Roman Nedela, Martin Skoviera, Na-Er Wang

Abstract


A dessin is a 2-cell embedding of a connected 2-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation- and colour-preserving automorphisms acts regularly on the edges. In this paper we study regular dessins whose underlying graph is a complete bipartite graph Km, n, called (m, n)-complete regular dessins. The purpose is to establish a rather surprising correspondence between (m, n)-complete regular dessins and pairs of skew-morphisms of cyclic groups. A skew-morphism of a finite group A is a bijection φ: A → A that satisfies the identity φ(xy) = φ(x)φπ(x)(y) for some function π: A → ℤ and fixes the neutral element of A. We show that every (m, n)-complete regular dessin


Keywords


regular dessin; bicyclic group; skew-morphism; graph embedding

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ISSN: 1855-3974

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