Tits alternative

Remarks

The two cases exclude each other, because subgroups of virtually solvable groups are virtually solvable and a non-abelian free group is not. Suppose $G$ is not virtually solvable. Tits embeds the relevant field into a suitable local field and finds two elements of $G$ that act on projective space with attracting and repelling points in general position. High powers of these elements then satisfy the ping-pong lemma, and so generate a free group of rank $2$.

Example: $\mathrm{SL}_2(\Z)$ is not virtually solvable, and indeed the matrices $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$ generate a free group of rank $2$. Example: the group of upper triangular invertible matrices is solvable, so it has no non-abelian free subgroup.

In positive characteristic, finite generation is needed. Let $\bar{\F}_p$ be an algebraic closure of $\F_p$. Every finitely generated subgroup of $\mathrm{GL}_2(\bar{\F}_p)$ is finite, so $\mathrm{GL}_2(\bar{\F}_p)$ has no free subgroup of rank $2$. But it is not virtually solvable, since it has the infinite simple group $\mathrm{PSL}_2(\bar{\F}_p)$ as a quotient of a subgroup.

Going deeper

For a student

The theorem gives strong consequences for linear groups. A finitely generated linear group is amenable if and only if it is virtually solvable, since groups with free subgroups are not amenable. Combined with the Milnor–Wolf theorem on finitely generated solvable groups, it shows that the growth of a finitely generated linear group is either polynomial or exponential. Groups outside linear groups are said to satisfy the Tits alternative when the same dichotomy holds for their subgroups; examples include hyperbolic groups and Out(F_n).