AU-Acylindricity may be viewed as generalizing the type of action a lattice enjoys on its ambient space. In a recent joint work with S. Balasubramanya, we extend the theory of acylindrically hyperbolic groups to the higher rank setting, using the theory of S-arithmetic lattices in semi-simple linear groups as motivation. This leads to an (imperfect) dictionary between the classical theory of algebraic groups and isometric actions on finite products of delta-hyperbolic spaces. In this talk we will focus on the techniques of this recent work and connect them to both classes of groups mentioned above.

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