In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted ${\displaystyle F\subseteq T\subseteq V}$, that were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, F is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group.

The Thompson groups, and F in particular, have a collection of unusual properties that have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented. The groups T and V are (rare) examples of infinite but finitely-presented simple groups. The group F is not simple but its derived subgroup [F,F] is and the quotient of F by its derived subgroup is the free abelian group of rank 2. F is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2.

It is conjectured that F is not amenable and hence a further counterexample to the long-standing but recently disproved von Neumann conjecture for finitely-presented groups: it is known that F is not elementary amenable.

Higman (1974) introduced an infinite family of finitely presented simple groups, including Thompson's group V as a special case.

A finite presentation of F is given by the following expression:

where [x,y] is the usual group theory commutator, xyx⁻¹y⁻¹.

Although F has a finite presentation with 2 generators and 2 relations, it is most easily and intuitively described by the infinite presentation:

The two presentations are related by x₀=A, x_n = A¹⁻ⁿBAⁿ⁻¹ for n>0.