Tensor product decompositions of II${}_1$ factors arising from extensions of amalgamated free product groups

In this paper we introduce a new family of icc groups $\mathrm{\Gamma }$ which satisfy the following product rigidity phenomenon, discovered in [DHI16] (see also [dSP17]): all tensor product decompositions of the II${}_1$ factor $L(\mathrm{\Gamma })$ arise only from the canonical direct product decompositions of the underlying group $\mathrm{\Gamma }$. 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.