Regular Cayley maps for cyclic groups

Abstract:

An orientably-regular map $M$ is a 2-cell embedding of a connected graph in a closed, orientable surface, with the property that the group $\mathrm {Aut}^\textrm {o}M$ of all orientation-preserving automorphisms acts transitively on the arcs of $M$. If $\mathrm {Aut}^\textrm {o}M$ contains a subgroup $A$ that acts regularly on the vertex set, then $M$ is called a regular Cayley map for $A$. In this paper, we answer a question of recent interest by providing a complete classification of the regular Cayley maps for the cyclic group $C_n$, for every possible order $n$. This is the first such classification for any infinite family of groups. The approach used is entirely algebraic and does not involve skew morphisms (but leads to a classification of all skew morphisms which have an orbit that is closed under inverses and generates the group). A key step is the use of a generalisation by Conder and Isaacs (2004) of Ito’s theorem on group factorisations, to help determine the isomorphism type of $\mathrm {Aut}^\textrm {o}M$. This group is shown to be a cyclic extension of a cyclic or dihedral group, dependent on $n$ and a single parameter $r$, which is a unit modulo $n$ that satisfies technical number-theoretic conditions. For each $n$, we enumerate all such $r$, and then in terms of $r$, we find the valence and covalence of the map, and determine whether or not the map is reflexible, and whether it has a representation as a balanced, anti-balanced or $t$-balanced regular Cayley map.