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

TL;DR

This paper investigates the distribution of finite quotient orders in finitely generated groups, revealing they have densities 0, 1/2, or 1, and applies these findings to analyze the density of orders in symmetric graphs.

Contribution

It characterizes the possible densities of finite quotient orders in finitely generated groups and applies these results to symmetric graphs.

Findings

Finite quotient orders have natural density 0, 1/2, or 1.

Characterization of when each density occurs.

Sets of orders of certain symmetric graphs have density 0.

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.