An embedding version of Rubin's theorem

TL;DR

This paper extends Rubin's theorem to an embedding context, demonstrating how group embeddings can induce specific equivariant maps, with applications to generalized Brin-Thompson groups.

Contribution

It introduces an embedding version of Rubin's theorem and explores its implications for group embeddings and equivariant maps, especially in the context of generalized Brin-Thompson groups.

Findings

Established an embedding version of Rubin's theorem.

Identified conditions under which group embeddings induce equivariant maps.

Provided examples involving generalized Brin-Thompson groups.

Abstract

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