Skew-morphisms of regular Cayley maps
Given a group \(G\), a generating set \(X=X^{-1}\) not containing \(1_G\), and a cyclic permutation \(p\) of \(X\), a Cayley map \(\text{CM}(G,X,p)\) is a 2-cell embedding of the Cayley graph of \(G\) with respect to \(X\) in an oriented surface \(S\), such that the cyclic permutation of the edges around any vertex, in the direction of the given orientation, agrees with \(p\). In addition, a Cayley map \(M\) is called regular if \(\Aut(M)\) acts regularly (i.e., freely and transitively) on the set of edges of \(M\). The authors prove some new results, and reprove some old ones, about regular Cayley maps using the notion of a skew-morphism: a permutation \(\phi:G \to G\) is called a skew-morphism of \(G\) if \(\phi(1_G)=1_G\), and if, for each \(g \in G\), there is a non-negative integer \(i=i(g)\) such that \(\phi(gh)=\phi(g)\phi^i(h)\) for all \(h\in G\). The main results are: (i) A Cayley map \(M= \text{CM}(G,X,p)\) is regular if and only if there exists a skew-morphism \(\phi\) of \(G\) whose restriction to \(X\) is \(p\), and (ii) a group \(G\) admits a regular Cayley map \(\text{CM}(G,X,p)\) if and only if there is a skew-morphism \(\phi\) of \(G\) which has a symmetric orbit \(X\) that generates \(G\).
scientific article; zbMATH DE number 1743020