Abstract: In joint work with Talia Fernos, we will introduce a new notion of non-positively curved groups : discrete countable groups acting (AU) - acylindrically on finite products of hyperbolic spaces. This is a demonstrably large class of groups inspired by the classical theory of (S-arithmetic) lattices and acylindrically hyperbolic groups. I will present our results that mimic the "rank-1" case .i.e. results similar to those proved in the case of acylindrically hyperbolic groups, including a "higher rank" version of the Tits Alternative. Up to virtual isomorphism, finitely generated groups in this class enjoy a strongly canonical product decomposition, allowing us to give a partial resolution to a recent conjecture of Sela. Time permitting, I will also talk about connections to lattice envelopes from the work of Bader-Caprace-Furman-Sisto.