Embeddings into Thompson's group v and coCF groups

It is shown in Lehnert and Schweitzer ('The co-word problem for the Higman-Thompson group is context-free', Bull. London Math. Soc. 39 (2007) 235-241) that R. Thompson's group $V$ is a co-context-free ($co\mathcal {CF}$) group, thus implying that all of its finitely generated subgroups are also $co\mathcal {CF}$ groups. Also, Lehnert shows in his thesis that $V$ embeds inside the $co\mathcal {CF}$ group ${\rm QAut}(\mathcal {T}-{2,c})$, which is a group of particular bijections on the vertices of an infinite binary 2-edge-coloured tree, and he conjectures that ${\rm QAut}(\mathcal {T}-{2,c})$ is a universal $co\mathcal {CF}$ group. We show that ${\rm QAut}(\mathcal {T}-{2,c})$ 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 $V$.