Hurwitz's regular map 37 of genus 7: A polyhedral realization

Abstract

A Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(g − 1) automorphisms, where g is the genus of the surface. The Hurwitz surface of least genus is the Klein quartic of genus 3. A polyhedral realization without self-intersections of Klein's quartic of genus 3 was found by E. Schulte and J. M. Wills in 1985. For the next possible genus of a Hurwitz surface, i.e., for the genus 7 case with 72 vertices, we provide a polyhedral realization without self-intersections. We also show a topological representation for which we have a corresponding model.

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

“…The automorphism group of a Hurwitz surface is a Hurwitz group, and it can be obtained in the following way. Consider the regular map {3, 7} on the hyperbolic plane, and take the group of its orientation-preserving automorphisms (following Coxeter, we denote this group by [3,7] + ). Now the automorphism groups of Hurwitz surfaces are precisely the non-trivial finite quotient groups of [3,7] + , by some suitable normal subgroup.…”

“…Consider the regular map {3, 7} on the hyperbolic plane, and take the group of its orientation-preserving automorphisms (following Coxeter, we denote this group by [3,7] + ). Now the automorphism groups of Hurwitz surfaces are precisely the non-trivial finite quotient groups of [3,7] + , by some suitable normal subgroup. The quotients of the regular map {3, 7} by the same normal subgroups are the Hurwitz maps {3, 7} r on the corresponding Hurwitz surfaces.…”

“…In this paper, we investigate the polyhedral realizability of the Hurwitz map {3, 7} 18 of genus 7 from a symmetry point of view. In spite of its high combinatorial symmetry of order 1008, the polyhedral realization of this regular map in [3] had no geometric symmetry at all. Although all former polyhedral realizations of regular maps must have socalled hidden symmetries, see the corresponding references in [3], they all do have some non-trivial geometric symmetries as well.…”

“…In spite of its high combinatorial symmetry of order 1008, the polyhedral realization of this regular map in [3] had no geometric symmetry at all. Although all former polyhedral realizations of regular maps must have socalled hidden symmetries, see the corresponding references in [3], they all do have some non-trivial geometric symmetries as well. So, a natural problem has remained to find a polyhedral realization of this regular map {3, 7} 18 with a non-trivial geometric symmetry.…”

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

“…When the reader is interested to get the automorphisms of regular maps expressed in terms of permutations, the software MAGMA is a very good tool, however it is not freely available. I have received from Marston Conder the following decisive input expression for MAGMA and Michael Cuntz printed for me the element T. However, the labeling of the vertices of the genus 7 Hurwitz surface that we obtain when we use this input in MAGMA is different from the one used in of [2] and in [1]. When we carry over the MAGMA labeling to that used in these former articles, we obtain the orientation reversing symmetry of order 2 via the permutation (2, 3)(4, 8)(5, 7)(9, 11) ( The vertices 1, 6, 10, 33, 40, 63, 67, 72 are fixed under this symmetry.…”