In a recent joint work with Sahana Balasubramanya (Lafayette College) we present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of δ-hyperbolic spaces with general type factors. The vastness of this class of groups is exhibited by recognizing that it contains, for example, S-arithmetic lattices with rank-1 factors, acylindrically hyperbolic groups, HHGs, groups with property (QT), and is closed under direct products, passing to (totally general type) subgroups, and finite index over-groups.

We show that, up to virtual isomorphism, finitely generated groups in this class enjoy a strongly canonical product decomposition. This semi-simple decomposition also descends to the outer-automorphism group, allowing us to give a partial resolution to a recent conjecture of Sela. We also show that the groups for which his conjecture is made is contained in the class of subdirect products in products of acylindrically hyperbolic groups.

In this talk, we will give an outline of our free vs abelian Tits’ Alternative for groups acting acylindrically on finite products of δ-hyperbolic spaces and discuss in more detail the semi-simple aspects of the theory.