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Density of quotient orders in groups and applications to locally-transitive graphs

Marston Conder, Gabriel Verret, Darius Young

Abstract

We prove that the set of orders of finite quotients of a finitely generated group has natural density 0, 1/2 or 1, and characterise when each of these cases occurs. We apply this to show that the sets of orders of various families of symmetric graphs have natural density 0.

Density of quotient orders in groups and applications to locally-transitive graphs

Abstract

We prove that the set of orders of finite quotients of a finitely generated group has natural density 0, 1/2 or 1, and characterise when each of these cases occurs. We apply this to show that the sets of orders of various families of symmetric graphs have natural density 0.

Paper Structure

This paper contains 8 sections, 17 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Number Theory
  4. Group Theory
  5. Main results for groups
  6. Applications to locally-transitive graphs
  7. Preliminaries
  8. Main results for locally-transitive graphs

Key Result

Theorem A

If $G$ is a finitely generated group, then exactly one of the following holds:

Theorems & Definitions (39)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • ...and 29 more