Random group
Encyclopedia
In mathematics
, random groups are certain group
s obtained by a probabilistic construction. They were introduced by Misha Gromov to answer questions such as "What does a typical group look like?"
It so happens that, once a precise definition is given, random groups satisfy some properties with very high probability, whereas other properties fail with very high probability. For instance, very probably random groups are hyperbolic group
s. In this sense, one can say that "most groups are hyperbolic".
Any group can by defined by a group presentation
involving generators and relations. For instance, the Abelian group
has a presentation with two generators 
and 
, and the relation 
, or equivalently 
. The main idea of random groups is to start with a fixed number of group generators 
, and imposing relations of the form 
where each 
is a random word involving the letters 
and their formal inverses 
. To specify a model of random groups is to specify a precise way in which 
, 
and the random relations 
are chosen.
Once the random relations
have been chosen, the resulting random group 
is defined in the standard way for group presentations, namely: 
is the quotient of the free group
with generators 
, by the normal subgroup 
generated by the relations 
seen as elements of 
:
and a number of relations 
are fixed. Fix an additional parameter 
(the length of the relations), which is typically taken very large.
Then, the model consists in choosing the relations
at random, uniformly and independently among all possible reduced words of length at most 
involving the letters 
and their formal inverses 
.
This model is especially interesting when the relation length
tends to infinity: with probability tending to 
as 
a random group in this model is hyperbolic
and satisfies other nice properties.
For instance, in the density model, the number of relations is allowed to grow with the length of the relations. Then there is a sharp "phase transition" phenomenon: if the number of relations is larger than some threshold, the random group "collapses" (because the relations allow to show that any word is equal to any other), whereas below the threshold the resulting random group is infinite and hyperbolic.
Constructions of random groups can also be twisted in specific ways to build group with particular properties. For instance, Gromov used this technique to build new groups that are counter-examples to an extension of the Baum-Connes conjecture.
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...
, random groups are certain group
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...
s obtained by a probabilistic construction. They were introduced by Misha Gromov to answer questions such as "What does a typical group look like?"
It so happens that, once a precise definition is given, random groups satisfy some properties with very high probability, whereas other properties fail with very high probability. For instance, very probably random groups are hyperbolic group
Hyperbolic group
In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. The notion of a hyperbolic group was introduced...
s. In this sense, one can say that "most groups are hyperbolic".
Definition
The definition of random groups depends on a probabilistic model on the set of possible groups. Various such probabilistic models yield different (but related) notions of random groups.Any group can by defined by a group presentation
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
involving generators and relations. For instance, the Abelian group
has a presentation with two generators and , and the relation , or equivalently . The main idea of random groups is to start with a fixed number of group generators , and imposing relations of the form where each is a random word involving the letters and their formal inverses . To specify a model of random groups is to specify a precise way in which , and the random relations are chosen.Once the random relations
have been chosen, the resulting random group is defined in the standard way for group presentations, namely: is the quotient of the free groupFree 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...
with generators , by the normal subgroup generated by the relations seen as elements of :
The few-relator model of random groups
The simplest model of random groups is the few-relator model. In this model, a number of generators and a number of relations are fixed. Fix an additional parameter (the length of the relations), which is typically taken very large.Then, the model consists in choosing the relations
at random, uniformly and independently among all possible reduced words of length at most involving the letters and their formal inverses .This model is especially interesting when the relation length
tends to infinity: with probability tending to as a random group in this model is hyperbolicHyperbolic group
In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. The notion of a hyperbolic group was introduced...
and satisfies other nice properties.
Further remarks
More refined models of random groups have been defined.For instance, in the density model, the number of relations is allowed to grow with the length of the relations. Then there is a sharp "phase transition" phenomenon: if the number of relations is larger than some threshold, the random group "collapses" (because the relations allow to show that any word is equal to any other), whereas below the threshold the resulting random group is infinite and hyperbolic.
Constructions of random groups can also be twisted in specific ways to build group with particular properties. For instance, Gromov used this technique to build new groups that are counter-examples to an extension of the Baum-Connes conjecture.