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 cocontext-context-free (coCF) group, thus implying that all of its finitely generated subgroups are also coCF groups. Also, Lehnert shows in his thesis that V embeds inside the coCF group QAut(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 QAut(T_2,c) is a universal coCF group. We show that QAut(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.