Tits alternative

The Tits alternative is an important theorem about the structure of finitely generated linear groups, proved by Jacques Tits and named after him. It says that a finitely generated group of matrices over a field is either virtually solvable, meaning it has a solvable subgroup of finite index, or it contains a non-abelian free subgroup. The result gives a sharp dichotomy: a linear group either behaves in a controlled, nearly solvable way or is large enough to contain a free group. One consequence is that a linear group is not amenable exactly when it contains a non-abelian free group, so the von Neumann conjecture, although false in general, holds for linear groups. The alternative is also an important ingredient in the proof of Gromov's theorem on groups of polynomial growth, which it essentially establishes for linear groups by reducing them to solvable groups. The original proof looks at the Zariski closure of the group in the general linear group and finishes with a ping-pong argument. In geometric group theory a group is said to satisfy the Tits alternative when each of its subgroups is virtually solvable or contains a non-abelian free subgroup; hyperbolic groups and mapping class groups are examples, while the Grigorchuk group and Thompson's group F are not.