An embedding version of Rubin's theorem

Rubin’s theorem asserts that if $\mathrm{\Gamma }\curvearrowright X$ and $\mathrm{\Delta }\curvearrowright Y$ are Rubin actions, then any group isomorphism $\mathrm{\Gamma }\cong \mathrm{\Delta }$ 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.