What are sources of finitely generated $\mathbb C$-linear groups that are not $\mathbb Z$-linear?

Recall that a group is *$R$-linear* if it is isomorphic to a subgroup of $GL(n,R)$ for some $n$, where $R$ is a ring.

I know only one source: any solvable $\mathbb Z$-linear group is polycyclic. For example, the Baumslag-Solitar group $B(1,2)$ is solvable, $\mathbb C$-linear, and it contains dyadic rationals, and therefore, is not polycyclic (abelian subgroups of polycyclic groups are finitely generated).

My personal motivation for the question is an attempt to digest recent applications of virtual Haken conjecture implying that many $3$-manifold groups are $\mathbb Z$-linear.