An embedding version of Rubin's theorem
Jan Gundelach
TL;DR
The paper generalizes Rubin's reconstruction theorem to embeddings by introducing Rubin embeddings and anchor maps that encode how an injective homomorphism $\Phi:\Gamma \to \Delta$ induces a spatial, $\Phi$-equivariant map between actions. It develops a machinery of localized subgroups and the saturated topology, defines a $\Phi$-equivariant order isomorphism $P$ on saturated regular supports, and derives a surjective anchor map $\rho:Y\to X$ via limits of ultrafilters, proving the embedding is captured by $\rho$. A key result is the equivalence between $\Phi$ being a Rubin embedding and the existence of a unique surjective anchor map that preserves regular supports, enabling spatial realization of embeddings. The framework supports canonical embeddings between generalized Brin-Thompson groups through table-coordinate constructions, illustrating practical instantiations like $\iota: V_2\hookrightarrow 2V_2$ with anchor $\pi_1$. This work provides a robust approach to realizing embeddings of topological full groups and related groupoid actions in terms of anchor maps and saturated-subgroup data. $
Abstract
Rubin's theorem asserts that if $Γ\curvearrowright X$ and $Δ\curvearrowright Y$ are Rubin actions, then any group isomorphism $Γ\cong Δ$ induces an equivariant homeomorphism $Y\cong X$. We provide an embedding version of Rubin's theorem highlighting group embeddings that induce a spatial equivariant map of a certain form. We further showcase instances of such embeddings between generalized Brin-Thompson groups.