Combinatorial conditions in residually finite groups. II (Q1802206)

The author continues his investigation of collapsing groups [for part I cf. the preceding review Zbl 0814.20022]. Let \(G\) be a group and let \(n\) be a positive integer. \(G\) is \(n\)-collapsing if for any set \(S\) of \(n\) elements in \(G\) we have \(| S^ n| < n^ n\). If \(G\) is \(n\)- collapsing for some \(n\), the author says that \(G\) is collapsing. \(G\) is strongly locally nilpotent if it generates a locally nilpotent variety. The author proves that a residually finite group is collapsing if and only if it is an extension of a strongly locally nilpotent group by a group of finite exponent. Also the author shows that there exist functions \(f\), \(g\) such that every finite group \(G\) which is \(n\)- collapsing possesses a nilpotent normal subgroup \(N\) satisfying (1) \(\text{exp}(G/N)\) divides \(f(n)\), and (2) every \(d\)-generated subgroup of \(N\) has class at most \(g(n,d)\).