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Chiral covers of regular maps of given type

Olivia Reade, Jozef Širáň

Abstract

With the help of the theory of holomorphic and anti-holomorphic differentials, G. A. Jones [Chiral covers of hypermaps, Ars Math. Contemp. 8 (2015), 425-431] proved that every regular hypermap of a non-spherical type is covered by an infinite number of orientably-regular but chiral hypermaps of the same type. We present a different proof of the same result for regular maps, based on parallel products of maps and existence of chiral maps of a given hyperbolic type with a symmetric or an alternating automorphism group.

Chiral covers of regular maps of given type

Abstract

With the help of the theory of holomorphic and anti-holomorphic differentials, G. A. Jones [Chiral covers of hypermaps, Ars Math. Contemp. 8 (2015), 425-431] proved that every regular hypermap of a non-spherical type is covered by an infinite number of orientably-regular but chiral hypermaps of the same type. We present a different proof of the same result for regular maps, based on parallel products of maps and existence of chiral maps of a given hyperbolic type with a symmetric or an alternating automorphism group.
Paper Structure (5 sections, 3 theorems, 7 equations)

This paper contains 5 sections, 3 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Orientably-regular maps: basic facts revisited
  3. Exceptional orientably-regular maps
  4. Parallel product of groups: a special case
  5. The result

Key Result

Lemma 1

Let $M={\rm Map}(G;x,y)$ be an exceptional orientably-regular map of type $\{m,n\}$ for some even $n$ and for some subgroup $\Gamma\,\triangleleft_{\,2}\, \Delta(m,n,2)$, of type A or B. Let $H\, \triangleleft_{\,2}\, G$ be the image of the restriction of the natural epimorphism $\Delta\to G$ to $\G

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • Theorem 1