We say that a group $G$ satisfies a law if there exists a (nontrivial) word $w \in \mathbb{F}_n$ such that $w(g_1,\dots,g_n)=1$ for every $g_1,\dots, g_n \in G$. For example, any abelian group satisfies the law $[r,s]=1$, and more generally, any metabelian group satisfies the law $[[r,s],[t,u]]=1$; contrariwise, a non abelian free group (and a fortiori any limit group) does not satisfy a law.
I find the notion interesting, but I did not find a lot of references on the subjet, except some papers dealing with the link between satisfying a law and not containing a non abelian free group (inspired by Tits' alternative).
So my (pretty vague) question is: Can we meet this notion in recent works? Is it a subset considered as interesting?