The semi-simple theory of higher rank acylindricity

TL;DR

This paper introduces a new class of non-positively curved groups acting acylindrically on hyperbolic spaces, establishing a canonical product decomposition and exploring structural properties, including implications for automorphism groups and lattice theory.

Contribution

It defines a broad new class of groups with acylindrical actions, proves a semi-simple product decomposition, and connects to conjectures and structure results in geometric group theory.

Findings

Finitely generated groups in this class have a canonical product decomposition.

The decomposition extends to outer automorphism groups, addressing Sela's conjecture.

The class includes diverse groups like S-arithmetic lattices and acylindrically hyperbolic groups.

Abstract

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. Inspired by the classical theory of ($S$-arithmetic) lattices and the flourishing theory of acylindrically hyperbolic 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 develop various structure results including a free vs abelian Tits' Alternative, and connections to lattice envelopes. Along the way we give representation-theoretic proofs of various results about acylindricity -- some methods are new even in the rank-1 setting. 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.