Complete regular dessins and skew-morphisms of cyclic groups

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 $K_{m,n}$, 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 $\phi :A\to A$ that satisfies the identity $\phi (xy)=\phi (x){\phi }^{\pi (x)}(y)$ for some function $\pi :A\to \mathbb{Z}$ and fixes the neutral element of~$A$. We show that every $(m,n)$-complete regular dessin $\mathcal{D}$ determines a pair of reciprocal skew-morphisms of the cyclic groups ${\mathbb{Z}}_n$ and ${\mathbb{Z}}_m$. Conversely, $\mathcal{D}$ can be reconstructed from such a reciprocal pair. As a consequence, we prove that complete regular dessins, exact bicyclic groups with a distinguished pair of generators, and pairs of reciprocal skew-morphisms of cyclic groups are all in one-to-one correspondence. Finally, we apply the main result to determining all pairs of integers $m$ and $n$ for which there exists, up to interchange of colours, exactly one $(m,n)$-complete regular dessin. We show that the latter occurs precisely when every group expressible as a product of cyclic groups of order $m$ and $n$ is abelian, which eventually comes down to the condition $\gcd (m,\varphi (n))=\gcd (\varphi (m),n)=1$, where $\varphi$ is Euler's totient function.

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