Acylindricity may be viewed as a generalization of being a uniform lattice in a locally compact second countable group. The theory of acylindrical actions on hyperbolic spaces has seen an explosion in recent years. Trees are of course examples of hyperbolic spaces, and by considering products, we start to see new and interesting behaviors that are not present in rank-1, such as the simple Burger-Mozes-Wise lattices, or Bestvina-Brady kernels.

In a joint work with S. Balasubramanya we introduce a new class of nonpositively curved groups. Viewing the theory of S-arithmetic semi-simple lattices as inspiration, we extend the theory of acylindricity to higher rank and consider finite products of $\delta$-hyperbolic spaces. The category is closed under products, subgroups, and finite index over-groups. Weakening acylindricity to AU-acylindricity (i.e. acylindricity of Ambiguous Uniformity) the theory captures all $S$-arithmetic semi-simple lattices with rank-1 factors, acylindrically hyperbolic groups, HHGs, and many others. In this talk, we will discuss structure theorems such as the Tits’ Alternative. This structure allows us to give a partial resolution to a conjecture by Sela.