Introduction

Thompson constructed a finitely presented group now known as V as an early example of a finitely presented infinite simple group. The group V contains a remarkable variety of subgroups, such as the finitary infinite permutation group S∞, and hence all (countable locally) finite groups, finitely generated free groups, finitely generated abelian groups, Houghton's groups, copies of Thompson's groups F, T and V, and many of their generalizations, such as the groups G n,r constructed by Higman [9]. Moreover, the class of subgroups of V is closed under direct products and restricted wreath products with finite or infinite cyclic top group.

In this short survey, we summarize the development of properties of V focusing on those which prohibit various groups from occurring as subgroups of V.

Thompson's group V has many descriptions. Here, we simply recall that V is the group of right-continuous bijections from the unit interval [0, 1] to itself, which map dyadic rational numbers to dyadic rational numbers, which are differentiable except at finitely many dyadic rational numbers, and with slopes, when defined, integer powers of 2. The elements of this group can be described by reduced tree pair diagrams of the type (S, T, π) where π is a bijection between the leaves of the two finite rooted binary trees S and T.

Higman [9] gave a different description of V, which he denoted as G2,1 in a family of groups generalizing V.

Obstructions

Higman [9] described several important properties of V which can serve as obstructions to subgroups occurring in V.

Theorem 2.1 ([9])An element of infinite order in V has only finitely many roots.

This prevents all Baumslag-Solitar groups Bm,n = <a, b | anb = bam> from occurring as subgroups of V, if m properly divides n; see [13].