The growth of residually soluble groups

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