Tits Alternative in groups with proper product actions on proper Gromov-hyperbolic spaces

In this paper, we study groups with property (PPH), i.e., there exist finitely many proper Gromov-hyperbolic spaces $X_1,\dots ,X_l$ on which $G$ acts cocompactly such that the diagonal action of $G$ on the ${\mathrm{\ell }}^1$-product ${\prod }_{i=1}^l{X}_i$ is proper. We show that any finitely generated subgroup of a finitely generated group with property (PPH) either is amenable or contains $F_2$. Furthermore, we study groups with property (PPT), i.e., groups with property (PPH) so that $X_1,\cdots ,X_l$ are all proper quasi-trees. We show that any finitely generated subgroup of a finitely generated group with property (PPT) either is virtually (locally-finite)-by-${\mathbb{Z}}^n$ or contains $F_2$. Additionally, we establish that for a non-elementary hyperbolic group $G$, $G$ admits a proper diagonal action on a finite product of regular trees if and only if $G$ has property (PPT). This result transforms a question posed by Button \cite{But19} into the problem of whether every non-elementary hyperbolic group has property (PPT).