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 φ: A → A that satisfies the identity φ(xy) = φ(x)φ^π(x)(y) for some function π: A → ℤ and fixes the neutral element of A. We show that every (m, n)-complete regular dessin D determines a pair of reciprocal skew-morphisms of the cyclic groups ℤ_n and ℤ_m. Conversely, 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 a 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 isomorphism class of (m, n)-complete regular dessins. 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, ϕ(n)) = gcd (ϕ(m), n) = 1, where ϕ is Euler’s totient function.