Tits alternative

In mathematics, the Tits alternative, named after Jacques Tits, is an important theorem about the structure of finitely generated linear groups.

Statement

The theorem, proven by Tits, is stated as follows.

A linear group is not amenable if and only if it contains a non-abelian free group.
The Tits alternative is an important ingredient in the proof of Gromov's theorem on [groups of polynomial growth]. In fact the alternative essentially establishes the result for linear groups.

In geometric group theory, a group G is said to satisfy the Tits alternative if for every subgroup H of G either H is virtually solvable or H contains a nonabelian free subgroup.
Examples of groups satisfying the Tits alternative which are either not linear, or at least not known to be linear, are:

Examples of groups not satisfying the Tits alternative are:

The proof of the original Tits alternative is by looking at the Zariski closure of in. If it is solvable then the group is solvable. Otherwise one looks at the image of in the Levi component. If it is noncompact then a ping-pong argument finishes the proof. If it is compact then either all eigenvalues of elements in the image of are roots of unity and then the image is finite, or one can find an embedding of in which one can apply the ping-pong strategy.
Note that the proof of all generalisations above also rests on a ping-pong argument.