Hurwitz Map

A Hurwitz map is an orientably regular map of type . Its dual, of type , is also often called a Hurwitz map. A regular map is orientably regular if it lies on an orientable surface and its orientation-preserving automorphism group acts transitively on its graph arcs (Conder and Melekoglu 2017). In the convention, every face is a triangle and seven faces meet at every vertex (Conder and Melekoglu 2017).

Every Hurwitz map is carried by a compact Riemann surface of genus whose orientation-preserving automorphism group has the maximum possible group order , and conversely every such surface carries a Hurwitz map. The corresponding automorphism group is a quotient group of the triangle group (Hurwitz 1893, Conder and Melekoglu 2017).

If the map has vertices, edges, and faces, then . Together with the Euler characteristic relation , this gives

Its orientation-preserving automorphism group therefore has group order . If the map is reflexible, its full automorphism group has group order .

The smallest Hurwitz map has genus 3 and is carried by the Klein quartic. The Hurwitz map of genus 7 lies on the Fricke-Macbeath curve and has the Fricke-Macbeath graph as its 1-skeleton. It has 72 vertices, 252 edges, and 168 faces. Its full automorphism group has group order 1008 (Bokowski and Cuntz 2018).