---
tag: 4J05N6H5NX
kind: concept
name: Random group
msc: [20P05, 20F65, 20F67]
sources:
  - url: https://en.wikipedia.org/wiki/Random_group
same_as:
  wikidata: Q7291971
  wikipedia: Random group
---

## Summary

A random group is a group given by a [[presentation|0JGEQXQ4AC]] whose relators are chosen at random.
The random model has a size parameter that tends to infinity.
One then asks which properties hold with probability tending to $1$.
Gromov's density model has a sharp threshold at density $\frac{1}{2}$.
Below it, random groups are infinite and [[hyperbolic|JWJ8YPR7EB]]; above it, they are trivial or of order $2$.
Random groups show what a typical finitely presented group looks like, and they provide examples of groups with unusual combinations of properties.

## Explanations

### L1 human

Choose a few letters as generators and write down many random words in them as relations.
The group you get is a random group.
With few relations compared to the number of possible words, the group is almost surely infinite and looks like a tree at large scales.
With too many relations, they collapse everything, and the group is almost surely trivial or has two elements.