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 $(\log \gamma(n))/ n^{1/4} \to 0$ 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.