# Applications of Tits' alternative in algebraic number theory

I have recently studying Tits' alternative. The theorem statement goes like the following:

Tits' alternative: Let $$G$$ be any finitely generated linear group over a field. Then one of the following is true,

$$(1)$$ $$G$$ contains a solvable normal subgroup of finite index,

$$(2)$$ $$G$$ contains a non-abelian free subgroup (of rank at least $$2$$).

I am in search of applications of this wonderful theorem in algebraic number theory. Any help, resources or reference will be appreciated. Thanks in advance.

• The proof of the Tits alternative really consists, assuming that $G$ is not virtually solvable, in constructing a free subgroup (rather, say, that assuming that $G$ contains no free subgroup, that $G$ is solvable). Hence the question is maybe what such free subgroups are useful to?
– YCor
Jul 13, 2020 at 0:25
• I don't know how it is applicable in algebraic number theory but it is a very important result in Geometric group theory, a branch deals with the study of finitely generated infinite groups using geometry. Tit's alternative is one of the key step used in proving one of the famous theorem of Gromov's which says that a finitely generated group has polynomial growth iff it has nilpotent subgroup of finite index. Jul 13, 2020 at 6:35