Random group

Summary

A random group is a group given by a presentation 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; 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.