Polynomial isoperimetric inequality for groups with function Δ(n) bounded by

We show that a polynomial isoperimetric inequality is true in ‘almost every group’, in some sense. Namely, we consider the function Δ(n),n Z,n>0, introduced, for a finitely generated group, by Robert Gilman, who proved that, for any group, Δ(n) n/3 , and that a slightly stricter condition Δ(n)< n/3 implies that the group is finitely presented and satisfies exponential isoperimetric inequality. In this paper, we show that the asymptotic condition , where >0 is arbitrarily small but fixed, implies polynomial isoperimetric inequality.