On the Growth of Residually Soluble Groups

It is shown that, for finitely generated residually solublegroups, a condition weaker than polynomial growth guaranteesvirtual nilpotence. Let G be a residually soluble group havinga finite generating set X, and suppose that the number _X(n)of elements of G that are products of at most n elements ofX X^-1 satisfies _X(n) e⁽ⁿ⁾ for each n, where (n)/^(1/2)(ln n)^1/2}as n then G is virtually nilpotent.