A Kepler-Poinsot-type polyhedron of the genus 7 Hurwitz surface

Abstract

In 2017 a first polyhedral embedding of the genus 7 Hurwitz surface of type {3, 7}18 was found by M. Cuntz and the author. For all previously determined polyhedral embeddings of regular maps, there exist those with non-trivial geometric symmetries as well. The orientation-preserving combinatorial automorphism group of this regular map of Hurwitz is the projective special linear group PSL(2, 8). For its subgroups, their possible corresponding geometric polyhedral embeddings have been investigated by G. Gévay and the author in this volume. There is an additional symmetry of order 2 that reverses the orientation. For this symmetry with eight fixed points, this paper provides a Kepler–Poinsot-type polyhedron which realizes this symmetry together with two additional symmetries of order 2. This polyhedron might serve as a starting point for proving that a geometric symmetry of order 2 for an embedding cannot exist.