Thompson groups
Encyclopedia
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 group
s. 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 known that F is not elementary amenable. If F is not amenable, then it would be 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.
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, xn = A1−nBAn−1 for n>0.
s, and as the group of piecewise linear homeomorphism
s of the unit interval
that preserve orientation and whose non-differentiable points are dyadic rationals and whose slopes are all powers of 2. The group T is an analogous group of piecewise line homeomorphisms of the unit circle
; V, of the Cantor set
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, 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 group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
s. The group F is "just non-abelian" in the sense that it is not abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, but all its proper homomorphic images are abelian. F is totally ordered, has exponential growth
Growth rate (group theory)
In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length...
, and does not contain a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
isomorphic to the free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
of rank 2. It is known that F is not elementary amenable. If F is not amenable, then it would be another counterexample to the long-standing but recently disproved
von Neumann conjecture
Von Neumann conjecture
In mathematics, the von Neumann conjecture stated that a topological group G is not amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture was disproved in 1980....
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.
Presentations
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, xn = A1−nBAn−1 for n>0.
Other representations
The group F also has realisations in terms of operations on binary treeBinary tree
In computer science, a binary tree is a tree data structure in which each node has at most two child nodes, usually distinguished as "left" and "right". Nodes with children are parent nodes, and child nodes may contain references to their parents. Outside the tree, there is often a reference to...
s, and as the group of piecewise linear homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
s of the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
that preserve orientation and whose non-differentiable points are dyadic rationals and whose slopes are all powers of 2. The group T is an analogous group of piecewise line homeomorphisms of the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
; V, of the Cantor set
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
.