Polynomial isoperimetric inequality for groups with function \(\Delta(n)\) bounded by \(n(\tfrac13-\varepsilon)\). (Q1420628)

Robert Gilman has introduced (for a finitely generated group) the function \(\Delta(n)\), \(n\in\mathbb{Z}\), \(n>0\), and proved that, for any group, \(\Delta (n)\leq[n/3]\), and that the stricter condition \(\Delta(n)<[n/3]\) implies that the group is finitely presented and satisfies exponential isoperimetric inequality. In the paper under review, the author shows that the asymptotic condition \(\Delta(n)\leq n(\tfrac13-\varepsilon)\), where \(\varepsilon>0\) is arbitrarily small but fixed, implies polynomial isoperimetric inequality.