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.
"Surface links with free abelian groups." J. Math. Soc. Japan 66 (1) 247 - 256, January, 2014. https://doi.org/10.2969/jmsj/06610247