Acylindricity in Higher Rank, Part I : Fundamentals

We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of $\delta$-hyperbolic spaces with general type factors and associated subdirect products. This work is inspired by the classical theory of $S$-arithmetic lattices and the flourishing theory of acylindrically hyperbolic groups. In this paper - the first of three - we develop various fundamental results, explore elementary subgroups in higher rank, and exhibit a free vs abelian Tits Alternative. Along the way we give representation-theoretic proofs of various results about acylindricity -- some methods are new even in the rank-one setting. The vastness of this class of groups is exhibited by recognizing that it contains $S$-arithmetic lattices with rank-one factors, acylindrically hyperbolic groups, colorable HHGs, groups with property (QT), and enjoys robust stability properties.