Totally disconnected locally compact groups with just infinite locally normal subgroups

TL;DR

This paper characterizes totally disconnected, locally compact groups where every locally normal subgroup has an open direct factor, linking this property to being locally isomorphic to a product of just infinite profinite groups.

Contribution

It provides a new characterization of certain totally disconnected groups based on their local normal subgroups and relates this to existing structure theories.

Findings

Groups are locally isomorphic to finite products of just infinite profinite groups.

The property is equivalent to a specific local normal subgroup condition when the quasi-centre is trivial.

Structural features of these groups are derived using advanced tools from group theory.

Abstract

We obtain a characterization of totally disconnected, locally compact groups $G$ with the following property: given a locally normal subgroup $K$ of $G$, then there is an open subgroup of $K$ that is a direct factor of an open subgroup of $G$. This property is motivated by J. Wilson's structure theory of just infinite groups, and indeed, when $G$ has trivial quasi-centre, the condition turns out to be equivalent to the condition that $G$ is locally isomorphic to a finite direct product of just infinite profinite groups. In the latter situation we obtain some global structural features of $G$, building on an earlier result of Barnea--Ershov--Weigel and also using tools developed by P.-E. Caprace, G. Willis and the author for studying local structure in totally disconnected locally compact groups.