Abstract
Just infinite profinite groups have proved to be a very fascinating family of groups and a rich source of interesting examples in Group Theory.
For a long time after the classification of finite simple groups, just infinite groups where considered the next classifiable family of groups. This proved to be much harder than expected and one of the many obstacles to this classification was the existence of hereditarily just infinite groups.
This thesis is concerned with the study of a generalisation of a new family of hereditarily just infinite profinite groups which are not virtually prop introduced by John Wilson in 2010, the \emph{generalised Wilson groups}, GW groups for short. Even though these examples are important, only few properties are known.
We start this work with a short overview of the known properties of generalised Wilson groups. Then, generalising a result of Bondarenko, we show that GW groups are finitely generated and we manage to produce explicit generators for GW groups.
We then consider other generationrelated profinite properties such as lower rank and finite presentability. We show that some generalised Wilson groups are new examples of profinite groups with finite lower rank. Moreover, we show that the direct product of certain hereditarily just infinite groups of finite lower rank still has finite lower rank. On the other hand we show that ``most'' GW groups are not finitely presentable.
In later chapters we look more closely at the subgroup structure of generalised Wilson groups. In particular we prove an embedding theorem for finitely generated profinite groups with specified composition factors in GW groups with the same set of composition factors. We study subgroup growth functions for some GW groups. Then, we prove that these groups are new examples of profinite groups with complete Hausdorff dimension spectrum. Finally, we analyze which GW groups are selfsimilar.
For a long time after the classification of finite simple groups, just infinite groups where considered the next classifiable family of groups. This proved to be much harder than expected and one of the many obstacles to this classification was the existence of hereditarily just infinite groups.
This thesis is concerned with the study of a generalisation of a new family of hereditarily just infinite profinite groups which are not virtually prop introduced by John Wilson in 2010, the \emph{generalised Wilson groups}, GW groups for short. Even though these examples are important, only few properties are known.
We start this work with a short overview of the known properties of generalised Wilson groups. Then, generalising a result of Bondarenko, we show that GW groups are finitely generated and we manage to produce explicit generators for GW groups.
We then consider other generationrelated profinite properties such as lower rank and finite presentability. We show that some generalised Wilson groups are new examples of profinite groups with finite lower rank. Moreover, we show that the direct product of certain hereditarily just infinite groups of finite lower rank still has finite lower rank. On the other hand we show that ``most'' GW groups are not finitely presentable.
In later chapters we look more closely at the subgroup structure of generalised Wilson groups. In particular we prove an embedding theorem for finitely generated profinite groups with specified composition factors in GW groups with the same set of composition factors. We study subgroup growth functions for some GW groups. Then, we prove that these groups are new examples of profinite groups with complete Hausdorff dimension spectrum. Finally, we analyze which GW groups are selfsimilar.
Original language  English 

Qualification  Ph.D. 
Awarding Institution 

Supervisors/Advisors 

Award date  1 Nov 2015 
Publication status  Unpublished  2015 