Random group

In mathematics, random groups are certain groups 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 groups. 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 be defined by a group presentation involving generators and relations. For instance, the Abelian group formula_1 has a presentation with two generators formula_2 and formula_3, and the relation formula_4, or equivalently formula_5. The main idea of random groups is to start with a fixed number of group generators formula_6, and imposing relations of the form formula_7 where each formula_8 is a random word involving the letters formula_9 and their formal inverses formula_10. To specify a model of random groups is to specify a precise way in which formula_11, formula_12 and the random relations formula_8 are chosen.
Once the random relations formula_14 have been chosen, the resulting random group formula_15 is defined in the standard way for group presentations, namely: formula_15 is the quotient of the free group formula_17 with generators formula_6, by the normal subgroup formula_19 generated by the relations formula_20 seen as elements of formula_17:
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 formula_23 and a number of relations formula_24 are fixed. Fix an additional parameter formula_25 (the length of the relations), which is typically taken very large.
Then, the model consists in choosing the relations formula_20 at random, uniformly and independently among all possible reduced words of length at most formula_25 involving the letters formula_9 and their formal inverses formula_10.
This model is especially interesting when the relation length formula_25 tends to infinity: with probability tending to formula_31 as formula_32 a random group in this model is hyperbolic 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.