On strongly just infinite profinite branch groups

TL;DR

This paper explores the properties of strongly just infinite profinite branch groups, establishing their key characteristics, automatic continuity properties, and providing concrete examples including the profinite completion of the first Grigorchuk group.

Contribution

It proves the equivalence of several properties for these groups, verifies automatic continuity properties under mild conditions, and presents new examples including universal simple groups.

Findings

Equivalence of Bergman property, uncountable cofinality, and other conditions.

Verification of automatic continuity properties for certain groups.

Examples include the profinite completion of the first Grigorchuk group.

Abstract

For profinite branch groups, we first demonstrate the equivalence of the Bergman property, uncountable cofinality, Cayley boundedness, the countable index property, and the condition that every non-trivial normal subgroup is open; compact groups enjoying the last condition are called strongly just infinite. For strongly just infinite profinite branch groups with mild additional assumptions, we verify the invariant automatic continuity property and the locally compact automatic continuity property. Examples are then presented, including the profinite completion of the first Grigorchuk group. As an application, we show that many Burger-Mozes universal simple groups enjoy several automatic continuity properties.