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

Abstract

In memory of Branko Grünbaum.

Hole operations on Hurwitz maps

“…For example, hexagons of type (2, 4) 3 occur in the polyhedral realisation of Klein's map K of genus 3 (see Section 7) given by Schulte and Wills in [31]. On the other hand, the heuristic role of hexagons of type (2, 4, 4) 2 is emphasised in the investigation of 7-fold rotational symmetry of the Fricke-Macbeath map F of genus 7 (see Section 8) in [3,4]. Moreover, in this map the 3-holes, together with three suitable triangular faces, form generalised Petersen graphs of type GP(9, 3) (see Figure 4); the presence of such subgraphs in the underlying graph of F confirms that this map has no polyhedral embedding in E 3 with 9-fold symmetry [4].…”

“…For other representations, including various topological embeddings by Carlo Sequin and by Jarke van Wijk, as well as for topological embeddings of regular maps of large genus given by Polthier and Razafindrazaka, see [3] and the references therein. For polyhedral realisations of the Hurwitz surface S of genus 7, see [2,3,4]. A detailed summary of all the known polyhedral realizations of regular maps of genus g ≥ 2 can be found in [4].…”

Hole operations on Hurwitz maps

“…For example, hexagons of type (2, 4) 3 occur in the polyhedral realisation of Klein's map K of genus 3 (see Section 7) given by Schulte and Wills in [29]. On the other hand, the heuristic role of hexagons of type (2, 4, 4) 2 is emphasised in the investigation of 7-fold rotational symmetry of the Fricke-Macbeath map F of genus 7 (see Section 8) in [3,4]. Moreover, in this map the 3-holes, together with three suitable triangular faces, form generalised Petersen graphs of type GP(9, 3) (see Figure 4); the presence of such subgraphs in the underlying graph of F confirms that this map has no polyhedral embedding in E 3 with 9-fold symmetry [4].…”

“…The combinatorial structure of this map is shown in Figure 9. For its polyhedral realisations, see [3,4,2].…”

“…In order to work with this group G, let us represent the field F 8 of order 8 as F 2 [t]/(t 3 + t + 1), with its elements represented as polynomials in F 2 [t] of Figure 9: Combinatorial scheme of the Hurwitz map F of genus 7 (with vertex labels taken from [4]). A 3-hole is highlighted.…”