  We consider the problem of deciding if a group is the fundamental group of a
smooth connected complex quasi-projective (or projective) variety using
Alexander-based invariants. In particular, we solve the problem for large
families of Artin-Tits groups. We also study finiteness properties of such
groups and exhibit examples of hyperplane complements whose fundamental groups
satisfy $\text{F}_{k-1}$ but not $\text{F}_k$ for any $k$.
