Reflexible complete regular dessins and antibalanced skew morphisms of cyclic groups

Abstract

A skew morphism of a finite group A is a bijection φ on A fixing the identity element of A and for which there exists an integer-valued function π on A such that φ(ab) = φ(a)φ^π(a)(b), for all a, b ∈ A. In addition, if φ^− 1(a) = φ(a^− 1)^− 1, for all a ∈ A, then φ is called antibalanced. In this paper we develop a general theory of antibalanced skew morphisms and establish a one-to-one correspondence between reciprocal pairs of antibalanced skew morphisms of the cyclic additive groups and isomorphism classes of reflexible regular dessins with complete bipartite underlying graphs. As an application, reflexible complete regular dessins are classified.