On hereditarily just infinite profinite groups obtained via iterated wreath products. (Q254317)

A profinite group \(G\) is just infinite if it is infinite, and every non-trivial closed normal subgroup is open; \(G\) is hereditarily just infinite if in addition \(H\) is just infinite for every open subgroup \(H\) of \(G\). The first example of a family of non-(virtually pro-\(p\)) hereditarily just infinite profinite groups was introduced by \textit{J. S. Wilson} [J. Algebra 324, No. 2, 248-255 (2010; Zbl 1209.20028)].NEWLINENEWLINE The author studies a generalisation of Wilson groups, considering a special type of infinitely iterated wreath product of finite non-abelian simple groups, called generalized Wilson groups. All the generalized Wilson groups are hereditarily just infinite and not virtually pro-\(p\). The author proves that this family contains groups of finite lower rank, i.e. having a base for the neighbourhoods of the identity consisting of \(r\)-generated subgroups, for some integer \(r\). He also shows that many groups in the family of generalized Wilson groups are not topologically finitely presentable.