ON THE GROWTH OF RESIDUALLY SOLUBLE GROUPS

Abstract

It is shown that, for finitely generated residually soluble groups, a condition weaker than polynomial growth guarantees virtual nilpotence. Let $G$ be a residually soluble group having a finite generating set $X$, and suppose that the number $\ga_X(n)$ of elements of $G$ that are products of at most $n$ elements of $X\cup X^{-1}$ satisfies $\ga_X(n)\leqslant e^{\alpha(n)}$ for each $n$, where $\alpha(n)/e^{(1/2)(\ln n)^{1/2}}\to\infty$ as $n\to\infty$; then $G$ is virtually nilpotent.