  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 $V$.
