Ranks of subgroups in boundedly generated groups

TL;DR

This paper proves that infinite boundedly generated residually finite groups have an infinite chain of finite index subgroups with bounded ranks, and provides upper bounds on ranks of finite index subgroups, confirming a conjecture about rank gradients.

Contribution

It establishes a strong form of a conjecture on rank gradients in boundedly generated groups and a variant of Lubotzky's conjecture on ranks of subgroups in special linear groups.

Findings

Infinite residually finite boundedly generated groups have an infinite chain of finite index subgroups with bounded ranks.

Provides sublinear upper bounds on ranks of finite index subgroups.

Confirms a conjecture that the rank gradient of such groups is zero.

Abstract

We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of boundedly generated groups (examples which come close to achieving these bounds are presented). This proves a strong form of a conjecture of Abert, Jaikin-Zapirain, and Nikolov which asserts that the rank gradient of infinite boundedly generated residually finite groups is $0$. Furthermore, our first result establishes a variant of a conjecture of Lubotzky on the ranks of finite index subgroups of special linear groups over the integers, and is analogous to a result of Pyber and Segal for solvable groups.