The Tits alternative is an important ingredient[2] in the proof of Gromov's theorem on groups of polynomial growth.
In fact the alternative essentially establishes the result for linear groups (it reduces it to the case of solvable groups, which can be dealt with by elementary means).
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 (in some versions of the definition this condition is only required to be satisfied for all finitely generated subgroups of G).
are roots of unity and then the image is finite, or one can find an embedding of
Note that the proof of all generalisations above also rests on a ping-pong argument.