---
tag: C2ATM03215
kind: definition
name: Random group
msc: [20P05, 20F65, 20F67]
defines: 4J05N6H5NX
uses: [0KTW26C80J, 0JGEQXQ4AC, JWJ8YPR7EB, 0BBTH3YSZA]
sources:
  - url: https://en.wikipedia.org/wiki/Random_group
---

## Definition

Fix an integer $m \ge 2$ and a real number $d$ with $0 \le d \le 1$, the *density*.
Let $F_m$ be the [[free group|0KTW26C80J]] on $a_1, \dots, a_m$.
For each integer $\ell \ge 1$, let $C_\ell$ be the [[finite set|0BBTH3YSZA]] of cyclically reduced words of length $\ell$ in $F_m$.
Let $R$ be a subset of $C_\ell$ with $\floor{(2m - 1)^{d \ell}}$ elements, chosen uniformly at random among all such subsets.
A *random group at density $d$ and length $\ell$* is the group
$$G = \langle a_1, \dots, a_m \mid R \rangle.$$
A property holds *for random groups at density $d$* if the probability that $G$ has it tends to $1$ as $\ell \to \infty$.

## Remarks

This is Gromov's density model.
The number of cyclically reduced words of length $\ell$ grows like $(2m - 1)^\ell$, so the density $d$ measures the number of relators on a logarithmic scale.
Gromov's theorem: for $d < \frac{1}{2}$, random groups at density $d$ are infinite, torsion-free and [[hyperbolic|JWJ8YPR7EB]]; for $d > \frac{1}{2}$, random groups at density $d$ are trivial or of order $2$.

Example: at density $d = 0$, the set $R$ has exactly one element, so the model produces a random one-relator group.

Non-example: a single fixed group, such as $\Z^2 = \langle a, b \mid aba^{-1}b^{-1} \rangle$, is not "random" in itself.
The definition describes a sequence of probability distributions on presentations, and only properties that hold with probability tending to $1$ are meaningful.

Other models fix the number of relators and let their length grow (the few-relator model), or use relators of length $3$ (the triangular model).