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

Abstract

In 2017 a first selfintersection-free polyhedral realization of Hurwitz’s regular map {3, 7}18 of genus 7 was found by Michael Cuntz and the first author. For any regular map which had previously been realized as a polyhedron without self-intersections in 3-space, it was also possible to find such a polyhedron with nontrivial geometric symmetries. 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) of order 504 = 23 ⋅ 32 ⋅ 7. All non-trivial subgroups of PSL(2,8) are candidates for such a geometric symmetry. Using the GAP software for exploring the subgroup structure, we found that it is sufficient to consider only four cyclic subgroups whose order is 9, 7, 3, and 2, respectively. We prove that there are obstructions for selfintersection-free polyhedral realizations of the Hurwitz map {3, 7}18 of genus 7 with geometric rotational symmetries of order 9 or 3. We provide new small integer coordinates within the realization space known from 2017, which are also suitable for making a 3D-printed model. We present Kepler–Poinsot type realizations, both with 7-fold and with 3-fold rotational symmetry, the latter with integer coordinates.