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Regular Cayley maps for cyclic groups

Authors: Marston D.E. Conder and Thomas W. Tucker
Journal: Trans. Amer. Math. Soc. 366 (2014), 3585-3609
MSC (2010): Primary 05E18; Secondary 05C10, 20B25, 57M15
Published electronically: March 3, 2014
MathSciNet review: 3192608
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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}^{\rm o}M$ of all orientation-preserving automorphisms acts transitively on the arcs of $ M$. If $ \mathrm {Aut}^{\rm 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}^{\rm 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.

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Additional Information

Marston D.E. Conder
Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

Thomas W. Tucker
Affiliation: Department of Mathematics, Colgate University, Hamilton, New York 13346

Received by editor(s): March 2, 2011
Received by editor(s) in revised form: August 9, 2012
Published electronically: March 3, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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