In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, commonly denoted
, that were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the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 group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The d ...
s. The group ''F'' is not simple but its
derived subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
'F'',''F''is and the quotient of ''F'' by its derived subgroup is the
free abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
of rank 2. ''F'' is
totally ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
, has
exponential growth
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
, and does not contain a
subgroup
In group theory, a branch of mathematics, 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 ...
isomorphic to the
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
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.
introduced an infinite family of finitely presented simple groups, including Thompson's group ''V'' as a special case.
Presentations
A finite presentation of ''F'' is given by the following expression:
:
where
'x'',''y''is the usual group theory
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
, ''xyx''
−1''y''
−1.
Although ''F'' has a finite presentation with 2 generators and 2 relations,
it is most easily and intuitively described by the infinite presentation:
: