Abstract. Lehnert and Schweitzer show in [20] that R. Thompson's group V is a co-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-colored 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 BaumslagSolitar groups embed into V .

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Let X be a finite set such that |X| = n. Let Tn and Sn denote the transformation monoid and the symmetric group on n points, respectively. Given a ∈ Tn \ Sn, we say that a group G Sn is a-normalizing ifwhere a, G and g −1 ag | g ∈ G denote the subsemigroups of Tn generated by the sets {a} ∪ G and {g −1 ag | g ∈ G}, respectively. If G is a-normalizing for all a ∈ Tn \ Sn, then we say that G is normalizing.The goal of this paper is to classify the normalizing groups and hence answer a question of Levi, McAlister, and McFadden. The paper ends with a number of problems for experts in groups, semigroups and matrix theory.