Branch actions and the structure lattice

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 $T$.

💡 Research Summary

The paper establishes a canonical, (G)-equivariant isomorphism between the structure lattice (L(G)) of a branch group (G) and the Boolean algebra of clopen subsets of the boundary (\partial T) of a spherically homogeneous rooted tree (T) on which (G) acts faithfully. The classical result of J. S. Wilson (1971) showed that for a just‑infinite branch group the structure lattice is abstractly isomorphic to the Boolean algebra of clopen subsets of the Cantor set, but the isomorphism did not respect the natural action of (G). The authors generalise Wilson’s theorem to all branch groups, removing the “just‑infinite’’ hypothesis, and provide an explicit description of the isomorphism.

The construction proceeds as follows. For a subgroup (H\le G) define its support
\