Exponents of orientable maps

Abstract

We generalize the idea of reflexibility of a map on a surface by introducing certain integers as its ”exponents“. An exponent is any integer $e$ with the property that changing the cyclic permutation of edges around each vertex induced by the map to its $e$-th power gives rise to an isomorphic map. The exponents reduced modulo the least common multiple of the vertex valencies form an Abelian group, the exponent group $\mbox{Ex}\,(M)$ of the map $M$. Along with the automorphism group, the group in fact provides an additional measure of symmetry of ~$M$.

The paper is devoted to developing the fundamentals of the theory of exponent groups of maps. Motivation comes from the problem of classification of regular maps with a given underlying graph. To this end, we prove that the number of non-isomorphic regular maps (if any) with a given underlying graph and the same map automorphism group is $|{\Bbb Z}_n^*:\mbox{Ex}\,(M)|$, $n$ being the valency of $M$. We calculate the exponent groups for some interesting families of regular maps including complete maps. Special attention is paid to the problem of how the antipodality of a map is reflected by its exponent group. In the final section we discuss several open problems.

1991 Mathematics Subject Classification: 05C10, 05C25, 20F32.