Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The *Tits alternative* says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$.

I am interested in linear groups $G$ which do not contain $F_2 \times F_2$ and are not virtually solvable. Do these groups have a characterization of some sort? In particular, is this true for $SL(3,\Bbb Z)$?

For the motivation, let me mention that if $G$ contains $F_2 \times F_2$, thеn the *group membership problem* (decision problem whether a group generated by a finite set of elements contains a given element) is undecidable.

K. A. Mihailova, The occurrence problem for direct products of groups, *Mat. Sb.* **70** (1966), 241-251.