Thompson groups

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This page is about the infinite Thompson groups F, T and V. For the sporadic finite simple group Th see Thompson group (finite).

In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted F, T and V, which were first studied by the logician Richard Thompson in 1965. 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 which have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented. T and V are (rare) examples of infinite but finitely presented simple groups. The group F is "just non-abelian" in the sense that it is not abelian, but all its proper homomorphic images are abelian. F is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2. It is not presently known whether F is amenable, but it is known not to be elementary amenable. If it turns out not to be amenable, then it will provide another counterexample to the long-standing but recently disproved von Neumann conjecture for finitely presented groups, which suggested that a finitely presented group is amenable if and only if it does not contain a copy of the free group of rank 2.

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

\langle A,B \mid\ [AB^{-1},A^{-1}BA] = [AB^{-1},A^{-2}BA^{2}] = \mathrm{id} \rangle

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:

\langle x_0, x_1, x_2, \dots\ \mid\ x_k^{-1} x_n x_k = x_{n+1}\ \mathrm{for}\ k<n \rangle.

It also has realisations in terms of operations on binary trees, and as a group of piecewise linear homeomorphisms of the unit interval.

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