Orthogonal forms of Kac--Moody groups are acylindrically hyperbolic

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.