Orientably Regular Maps of Given Hyperbolic Type with No Non-trivial Exponents

Abstract

Given an orientable map ${\mathcal {M}}$, and an integer e relatively prime to the valency of ${\mathcal {M}}$, the eth rotational power ${\mathcal {M}}^e$ of ${\mathcal {M}}$ is the map formed by replacing the cyclic rotation of edges around each vertex with its eth power. If ${\mathcal {M}}$ and ${\mathcal {M}}^e$ are isomorphic, and the corresponding isomorphism preserves the orientation of the carrier surface, then we say that e is an exponent of ${\mathcal {M}}$.In this paper, we use canonical regular covers of maps to prove that for every given hyperbolic pair (k, m) there exists an orientably regular map of type $\{m, k\}$ with no non-trivial exponents. As an application we show that for every given hyperbolic pair (k, m) there exist infinitely many orientably regular maps of type $\{m, k\}$ with no non-trivial exponents, each with the property that the map and its dual have simple underlying graph.