Some embeddings between symmetric R. Thompson groups

Abstract

Let $m\le n\in \mathbb{N}$, and $G\le S_m$ and $H\le S_n$. In this article we find conditions enabling embeddings between the symmetric R. Thompson groups $V_m(G)$ and $V_n(H)$. When $n\equiv 1\mathrm{mod}(m-1)$ and under some other technical conditions we find an embedding of $V_n(H)$ in $V_m(G)$ via topological conjugation. With the same modular condition we also generalise a purely algebraic construction of Birget from 2019 to find a group $H\le S_m$ and an embedding of $V_m(G)$ in $V_n(H)$.

16 pages. The only significant changes are in the introduction, where it has been given a deeper discussion of the relevant history