  It is known that if a classical link group is a free abelian group, then its
rank is at most two. It is also known that a $k$-component 2-link group ($k>1$)
is not free abelian. In this paper, we give examples of $T^2$-links each of
whose link groups is a free abelian group of rank three or four. Concerning the
$T^2$-links of rank three, we determine the triple point numbers and we see
that their link types are infinitely many.
