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An orientably-regular map M is a 2-cell embedding of a finite connected graph in a closed orientable surface such that the group Aut ∘ M of orientation-preserving automorphisms of M acts transitively on the set of arcs. Such a map M is called a Cayley map for the finite group G if Aut ∘ M contains a subgroup, which is isomorphic to G and acts regularly on the set of vertices. Conder and Tucker (2014) classified the regular Cayley maps for finite cyclic groups, and obtain two two-parameter families M (n , r) , one for odd n and one for even n , where n is the order of the regular cyclic group and r is a positive integer satisfying certain arithmetical conditions. In this paper, we classify the regular Cayley maps for dihedral groups in the same fashion. Five two-parameter families M i (n , r) , 1 ≤ i ≤ 5 , are derived, where 2 n is the order of the regular dihedral group and r is an integer satisfying certain arithmetical conditions. For each map M i (n , r) , we determine its valence and covalence, and also describe the structure of the group Aut ∘ M i (n , r). Unlike the approach of Conder and Tucker, which is entirely algebraic, we follow the traditional combinatorial representation of Cayley maps, and use a combination of permutation group theoretical techniques, the method of quotient Cayley maps, and computations with skew morphisms. [ABSTRACT FROM AUTHOR]

A regular map M is a 2-cell embedding of a connected graph into an orientable surface such that the group of all orientation-preserving automorphisms of the embedding acts transitively on the set of all incident vertex-edge pairs called arcs. Such a map M is called a regular Cayley map for the finite group G if M is the embedding of a Cayley graph C ( G , S ) such that G induces a vertex-transitive group of map automorphisms preserving orientation. In addition, if there is an orientation-reversing automorphism, the map is called reflexible. In this paper, we classify all reflexible Cayley maps for dihedral groups. [ABSTRACT FROM AUTHOR]

A linear algebraic group G over a field K is called a Cayley K -group if it admits a Cayley map, i.e., a G -equivariant K -birational isomorphism between the group variety G and its Lie algebra. We classify real reductive algebraic groups of absolute rank 1 and 2 that are Cayley R -groups. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, properties of reflexible Cayley maps for abelian groups are investigated, and as a result, it is shown that a regular Cayley map of valency greater than 2 for a cyclic group is reflexible if and only if it is anti-balanced. [Copyright &y& Elsevier]

Abstract: It is shown that every connected vertex and edge transitive graph has a normal multicover that is a connected normal edge transitive Cayley graph. Moreover, every chiral or regular map has a normal cover that is a balanced chiral or regular Cayley map, respectively. As an application, a new family of half-transitive graphs is constructed as 2-fold covers of a family of 2-arc transitive graphs admitting Suzuki groups. [Copyright &y& Elsevier]