Embeddings into Thompson's group $V$ and $co\mathcal{C}\mathcal{F}$ groups

Abstract

Lehnert and Schweitzer show in [20] that R. Thompson's group $V$ is a co-context-free ($co\mathcal{C}\mathcal{F}$) group, thus implying that all of its finitely generated subgroups are also $co\mathcal{C}\mathcal{F}$ groups. Also, Lehnert shows in his thesis that $V$ embeds inside the $co\mathcal{C}\mathcal{F}$ group $\mathrm{QAut}({\mathcal{T}}_{2,c})$, which is a group of particular bijections on the vertices of an infinite binary $2$-edge-colored tree, and he conjectures that $\mathrm{QAut}({\mathcal{T}}_{2,c})$ is a universal $co\mathcal{C}\mathcal{F}$ group. We show that $\mathrm{QAut}({\mathcal{T}}_{2,c})$ embeds into $V$, and thus obtain a new form for Lehnert's conjecture. Following up on these ideas, we begin work to build a representation theory into R. Thompson's group $V$. In particular we classify precisely which Baumslag-Solitar groups embed into $V$.

15 pages, 2 figures; preliminary version posted at the request of some colleagues

References in corpus (1)

• Free products in R. Thompson's group V

Cited by in corpus (3)

• Infinite $\frac{3}{2}$-generated groups
• Quasi-Automorphism Groups of Type F-infinity
• On Thompson Groups for Ważewski Dendrites