We study a generalisation of the family of non-(virtually pro-$p$) hereditarily just infinite profinite groups introduced by J.\! S.\! Wilson in 2010. We prove that this family contains groups of finite lower rank. We also show that many groups in this family are not topologically finitely presentable.
Finite generation of iterated wreath products
Generation of iterated wreath products constructed from alternating, symmetric and cyclic groups
On hereditarily just infinite profinite groups with complete Hausdorff dimension spectrum
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