Orthogonal forms of Kac--Moody groups are acylindrically hyperbolic

TL;DR

This paper establishes conditions under which groups acting on certain geometric spaces are acylindrically hyperbolic, with applications to Kac--Moody groups, buildings, and graph products, expanding understanding of their geometric group properties.

Contribution

It provides new criteria for acylindrical hyperbolicity in groups acting on buildings and applies these to orthogonal forms of Kac--Moody groups and graph products.

Findings

Orthogonal forms of Kac--Moody groups are acylindrically hyperbolic.

Groups acting on irreducible non-spherical non-affine buildings can be acylindrically hyperbolic.

Certain classes of groups admit actions on buildings with specific properties.

Abstract

We give sufficient conditions for a group acting on a geodesic metric space to be acylindrically hyperbolic and mention various applications to groups acting on CAT($0$) spaces. We prove that a group acting on an irreducible non-spherical non-affine building is acylindrically hyperbolic provided there is a chamber with finite stabiliser whose orbit contains an apartment. Finally, we show that the following classes of groups admit an action on a building with those properties: orthogonal forms of Kac--Moody groups over arbitrary fields, and irreducible graph products of arbitrary groups - recovering a result of Minasyan--Osin.