The growth of residually soluble groups

TL;DR

This paper proves that finitely generated residually soluble groups with sufficiently slow growth are virtually nilpotent, confirming the Gap Conjecture for exponents less than 1/4 within this class and improving previous results.

Contribution

It establishes a new growth threshold for residually soluble groups to be virtually nilpotent, extending the validity of the Gap Conjecture to a larger exponent range.

Findings

Proves residually soluble groups with growth condition are virtually nilpotent.

Extends the Gap Conjecture validity to exponents less than 1/4.

Improves Wilson's previous exponent bound of 1/6.

Abstract

Building on work of Wilson, we show that if $G$ is a finitely generated residually soluble group whose growth function $\gamma$ satisfies $(\log \gamma(n))/ n^{1/4} \to 0$ as $n \to \infty$ then $G$ is virtually nilpotent. This shows that Grigorchuk's Gap Conjecture holds for all exponents $\beta < 1/4$ within the class of residually soluble groups (improving Wilson's exponent $1/6$). We also discuss stronger versions of the Gap Conjecture.