  In this paper we introduce a new family of icc groups $\Gamma$ which satisfy
the following product rigidity phenomenon, discovered in [DHI16] (see also
[dSP17]): all tensor product decompositions of the II$_1$ factor $L(\Gamma)$
arise only from the canonical direct product decompositions of the underlying
group $\Gamma$. Our groups are assembled from certain HNN-extensions and
amalgamated free products and include many remarkable groups studied throughout
mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes
groups, Higman group, various integral two-dimensional Cremona groups, etc. As
a consequence, we obtain several new examples of groups that give rise to prime
factors.
