Branch actions and the structure lattice

TL;DR

This paper generalizes Wilson's 1971 isomorphism between structure lattices of certain groups and Boolean algebras of clopen sets to the broader class of branch groups, establishing a canonical isomorphism for their actions on rooted trees.

Contribution

It extends Wilson's isomorphism to branch groups and shows a canonical G-equivariant isomorphism between structure lattices and boundary clopen sets.

Findings

Generalization of Wilson's isomorphism to branch groups

Existence of a canonical G-equivariant isomorphism for branch actions

Connection between structure lattices and boundary clopen sets

Abstract

J. S. Wilson proved in 1971 an isomorphism between the structural lattice associated to a group belonging to his second class of groups with every proper quotient finite and the Boolean algebra of clopen subsets of Cantor's ternary set. In this paper we generalize this isomorphism to the class of branch groups. Moreover, we show that for every faithful branch action of a group $G$ on a spherically homogeneous rooted tree $T$ there is a canonical $G$-equivariant isomorphism between the Boolean algebra associated with the structure lattice of $G$ and the Boolean algebra of clopen subsets of the boundary of $T$.