Maximal subgroups of the modular and other groups

TL;DR

This paper constructs uncountably many conjugacy classes of maximal subgroups in various modular and triangle groups, extending classical results with new methods involving planar maps and elliptic elements.

Contribution

It introduces a simple planar map-based construction to generate uncountably many maximal subgroups in triangle groups and extends existing results to Hecke and other groups.

Findings

Uncountably many conjugacy classes of nonparabolic maximal subgroups in triangle groups.

Construction of uncountably many maximal subgroups generated by elliptic elements.

Existence of uncountably many torsion-free maximal subgroups in Hecke groups.

Abstract

In 1933 B.~H.~Neumann constructed uncountably many subgroups of ${\rm SL}_2(\mathbb Z)$ which act regularly on the primitive elements of $\mathbb Z^2$. As pointed out by Magnus, their images in the modular group ${\rm PSL}_2(\mathbb Z)\cong C_3*C_2$ are maximal nonparabolic subgroups, that is, maximal with respect to containing no parabolic elements. We strengthen and extend this result by giving a simple construction using planar maps to show that for all integers $p\ge 3$, $q\ge 2$ the triangle group $\Gamma=\Delta(p,q,\infty)\cong C_p*C_q$ has uncountably many conjugacy classes of nonparabolic maximal subgroups. We also extend results of Tretkoff and of Brenner and Lyndon for the modular group by constructing uncountably many conjugacy classes of such subgroups of $\Gamma$ which do not arise from Neumann's original method. These maximal subgroups are all generated by elliptic elements, of finite order, but a similar construction yields uncountably many conjugacy classes of torsion-free maximal subgroups of the Hecke groups $C_p*C_2$ for odd $p\ge 3$. Finally, an adaptation of work of Conder yields uncountably many conjugacy classes of maximal subgroups of $\Delta(2,3,r)$ for all $r\ge 7$.