Branch actions and the structure lattice
Jorge Fariña-Asategui, Rostislav Grigorchuk
TL;DR
The paper extends Wilson's structural isomorphism from a restricted class to all branch groups by constructing a canonical $G$-equivariant isomorphism between the structure lattice $\mathcal{L}(G)$ and the Boolean algebra $\mathrm{Bool}(\partial T)$ of clopen boundary subsets. It provides an explicit description of the lattice as $H\mapsto \mathrm{Supp}(H)$ with $C\mapsto [\prod_{v\in C}\mathrm{rist}_G(v)]$, and proves that the $G$-action on the Stone space of $\mathcal{L}(G)$ matches the natural action on $\partial T$, thereby bridging algebraic and dynamical aspects of branch groups. The results yield a concrete, lattice-level analog of Hardy's structure graph description, giving a complete, computable image of $\mathcal{L}(G)$ in terms of boundary clopens. This connects the internal subgroup lattice to external dynamical action, enabling explicit analysis of branch actions and their associated lattices.
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 $T$.