On generation problems in generalised Wilson type groups

TL;DR

This paper investigates a family of hereditarily just infinite profinite groups formed via iterated wreath products, providing explicit generators, analyzing their lower rank, and exploring their finite presentation properties.

Contribution

It introduces explicit generators for these groups, studies their lower rank, and examines the possibility of finite profinite presentations, advancing understanding of their structural properties.

Findings

Explicit generators identified for several cases

Lower rank of the groups determined

Conditions for finite profinite presentation discussed

Abstract

We study a family of hereditarily just infinite profinite groups obtained by iterated wreath products introduced by J. Wilson in 2010. We find explicit generators for this family in a number of cases using combinatorial methods. We then discuss determination of the lower rank for this family. Finally we examine whether the groups in this family admit a finite profinite presentation.