The growth of residually soluble groups

Abstract

Building on work of Wilson, we show that if $G$ is a finitely generated residually soluble group whose growth function ${\hat{I}}^3$ satisfies $(\log {\hat{I}}^3(n))/n^{1/4}\to 0$ as $n\to \infty$ then $G$ is virtually nilpotent. This shows that Grigorchuk's Gap Conjecture holds for all exponents ${\hat{I}}^2<1/4$ within the class of residually soluble groups (improving Wilson's exponent $1/6$). We also discuss stronger versions of the Gap Conjecture.

16 pp