On (hereditarily) just infinite profinite groups that are not virtually pro-p

TL;DR

This paper develops criteria to characterize when certain profinite groups are just infinite or hereditarily just infinite, especially focusing on those not virtually pro-p, expanding understanding of their structural properties.

Contribution

It provides new criteria for identifying just infinite and hereditarily just infinite properties in profinite groups that are not virtually pro-p.

Findings

Criteria for just infinite groups without being virtually pro-p

Criteria for hereditarily just infinite groups in this setting

Extension of Wilson's results to broader classes of profinite groups

Abstract

A profinite group G is just infinite if every non-trivial closed normal subgroup of G is of finite index, and hereditarily just infinite if every open subgroup is just infinite. Hereditarily just infinite profinite groups need not be virtually pro-p, as shown in a recent paper of Wilson. The same paper gives a criterion on an inverse system of finite groups that is sufficient to ensure the limit is either virtually abelian or hereditarily just infinite. We give criteria of a similar nature that characterise the just infinite and hereditarily just infinite properties under the assumption that G is not virtually pro-p.