Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Chiral maps of given hyperbolic type on $A_k$

Olivia Reade

Abstract

This paper proves the existence of a chiral map with alternating automorphism group for every hyperbolic type. We present a set of constructions using permutations for when at least one parameter is even, and call on previously known results for when both the valency and the face-length are odd.

Chiral maps of given hyperbolic type on $A_k$

Abstract

This paper proves the existence of a chiral map with alternating automorphism group for every hyperbolic type. We present a set of constructions using permutations for when at least one parameter is even, and call on previously known results for when both the valency and the face-length are odd.
Paper Structure (10 sections, 20 theorems, 10 figures, 1 table)

This paper contains 10 sections, 20 theorems, 10 figures, 1 table.

Table of Contents

  1. Introduction and context
  2. Background
  3. Chiral maps
  4. Permutation groups and primitivity
  5. Recognising chirality in permutation diagrams
  6. When $m$ is even and $n$ is odd
  7. Even $m < n$ odd
  8. Even $m > n$ odd
  9. When both $m$ and $n$ are even
  10. Proof of Theorem \ref{['thm:Thispaper']}

Key Result

Theorem 1

Given a hyperbolic type $\{m ,n \}$, there exists a chiral map of that type with alternating automorphism group $A_k$, for some degree $k$.

Figures (10)

  • Figure 1: Construction \ref{['con:noddmis4']} when $i=-1$ and $i=1$ respectively
  • Figure 2: Construction \ref{['con:noddmis6']} when $i=-1$ and $i=1$ respectively
  • Figure 3: Construction \ref{['con:noddmis8']} when $i=-1$ and $i=1$ respectively
  • Figure 4: Construction \ref{['con:noddmisgeq10']} when $i=-1$ and $i=1$ respectively
  • Figure 5: Construction \ref{['con:ngeq7']} when odd $n \geq 7$
  • ...and 5 more figures

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 25 more