Embeddings into Thompson's group V and coCF groups

Journal of the London Mathematical Society, 94(2) (2016), 583-597
http://arxiv.org/abs/1312.1855 and http://onlinelibrary.wiley.com/doi/10.1112/jlms/jdw044/epdf

Lehnert and Schweitzer show in [20] 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 $\mathrm{QAut}(\mathcal{T}_{2,c})$, 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 $co\mathcal{CF}$ 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 $V$.