Quotient order density of ordinary triangle groups

TL;DR

This paper proves that the set of finite quotient orders of any ordinary triangle group has natural density zero, using methods from group theory and number theory, specifically inspired by Bertram's theorem and the Turan-Kubilius inequality.

Contribution

It establishes that the natural density of finite quotient orders of ordinary triangle groups is zero, a result previously conjectured but not proven.

Findings

The density of finite quotient orders is zero for all ordinary triangle groups.

The proof combines group theory and number theory techniques.

The result answers a longstanding open question in the field.

Abstract

In this paper, we show that the natural density (among the positive integers) of the set of orders of the finite quotients of any ordinary triangle group is zero. This is achieved using methods inspired by a 1976 theorem of Bertram on large cyclic subgroups of finite groups and an application of the Turan-Kubilius inequality from asymptotic number theory. The question of the natural density of the set of finite quotient orders of triangle groups was first considered by Larson, then May and Zimmerman and then Tucker, who brought it to the author's attention.