A refined Hurwitz theorem for embeddings of irredundant Cayley graphs
\textit{Hurwitz}'s theorem [Math. Ann. 41, 403-442 (1892; JFM 24.0380.02)] states that the order of any finite group acting on a surface of genus \(\gamma >1\) has order at most 168(\(\gamma\)-1). In this paper similar results are obtained for Cayley graphs imbedded in a surface of genus \(\gamma\) ; these results have important implications for the classification of groups of low genus (particularly genus one and two) and the number of Cayley graphs of a given genus. In particular, a Cayley graph C(A,X) for a finite group A generated by \(X\subseteq A\) is said to be irredundant if X has no redundant generators. The genus (respectively characteristic) of a group A is the minimum genus (maximum characteristic) over all Cayley graphs C(A,X) for A, where the genus (respectively characteristic) of a graph G is the minimum genus over all orientable surfaces (maximum characteristic over all surfaces, orientable and nonorientable) into which G can be imbedded. Then the author summarizes some of the results of this paper (pertaining to the genus of C(A,X), not necessarily of A) in the following theorem: Let C(A,X) be an irredundant Cayley graph of genus \(\gamma\) and characteristic \(\chi\). If \(\gamma >1\), then \(\chi <0\). If \(| A| >12| \chi |,\) then \(| X| =2\) or 3 and if \(| X| =3\) then every element of X is an involution; moreover, X is a generating set of one of three types (depending upon the number of involutions - only 0, 1, and 3 are possible - and specified in the paper). In particular, \(| A| \leq 84| \chi |\) and the number of irredundant Cayley graphs of characteristic \(\chi <0\) or genus \(\gamma >1\) is finite. An example is given to show that irredundancy is necessary, when \(\gamma =2\).
scientific article; zbMATH DE number 3877216