On polyhedral realizations of Hurwitz's regular map {3, 7}_18 of genus 7 with geometric symmetries

Keywords: Hurwitz surface, regular map, Kepler-Poinsot type polyhedron


In 2017 a first selfintersection-free polyhedral realization of Hurwitz's regular map {3, 7}_{18} was found by Michael Cuntz and the first author. For all previously determined selfintersection-free polyhedral realizations of regular maps, there exist those with non-trivial geometric symmetries
as well. So it is natural to ask of whether we can find for the above-mentioned regular map a corresponding version with some non-trivial geometric symmetry. The orientation-preserving combinatorial automorphism group of this Hurwitz map is the projective special linear group PSL(2, 8). All non-trivial subgroups of PSL(2, 8) are candidates for such a geometric symmetry. Each subgroup has among their generators one of only 4 cyclic subgroups, with orders 9, 7, 3, or 2. We prove that there are obstructions for selfintersection-free polyhedral realizations of the Hurwitz map {3, 7}_{18} with geometric symmetries for these subgroups of order 9 or 3, and for the subgroup of order 2, when we assume a reflection in a point or in a plane. We provide small integer coordinates within the known realization space of the new realization. We present Kepler-Poinsot type realizations, both with 7-fold and with 3-fold rotational symmetry, the latter with integer coordinates.

Bled 2019 – Polytopes, Configurations, and Symmetries (Grünbaum Issue)