The semi-simple theory of higher rank acylindricity
Abstract
We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of -hyperbolic spaces with general type factors. Inspired by the classical theory of (-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, -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.
Keywords
Cite
@article{arxiv.2407.04838,
title = {The semi-simple theory of higher rank acylindricity},
author = {Sahana Balasubramanya and Talia Fernos},
journal= {arXiv preprint arXiv:2407.04838},
year = {2025}
}
Comments
An unproven claim and changes to the existing literature affected our work. All of the results herein are sill true, but addressing the changes required many additional pages of writing. So we have split this work into two parts for readability. Part I is available as arXiv:2512.21936 and Part II will be released shortly. For details of the changes, we refer the reader to Section 2 of Part 1