Embeddings into Thompson's group <i>V</i> and coCF groups

Abstract

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 ( c o CF ) group, thus implying that all of its finitely generated subgroups are also c o CF groups. Also, Lehnert shows in his thesis that V embeds inside the c o CF 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 c o CF 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.