The growth of residually soluble groups

TL;DR

This work advances the Gap Conjecture for finitely generated residually soluble groups by proving that if the growth function satisfies $(\log \gamma_X(n))/n^{1/4} \to 0$, then the group is virtually nilpotent, thereby establishing Gap$(\beta)$ for all $\beta<1/4$ in this class and improving previous exponent bounds. The authors develop a framework combining Milnor's lemma, a generalized quantitative Milnor lemma, and a new invariant called the modified derived length $\mu(G)$, together with Suprunenko's structure theorem and Newman bounds for soluble linear groups, to tightly control the growth via subnormal series. They prove explicit bounds for $\mu(G)$ in both permutation and linear group contexts, leading to a strengthened Gap bound of $\operatorname{Gap}(1/4.16)$ and ultimately $\operatorname{Gap}(1/4-\varepsilon)$ for residually soluble groups. The paper also discusses stronger conjectures Gap$^*(\beta)$ and Gap$^{**}(\beta)$ and outlines remaining questions and potential uniform growth phenomena, highlighting the broader impact on the growth-structure dichotomy in soluble groups and related areas.

Abstract

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

Theorems & Definitions (34)

• Conjecture : $\operatorname{Gap}(\beta)$
• Theorem 1.1: $\operatorname{Gap}(1/4-\epsilon)$ for residually soluble groups
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• proof
• Corollary 2.4
• Lemma 2.5
• proof
• Theorem 2.6