Orthogonal forms of Kac–Moody groups are acylindrically hyperbolic

Caprace, Pierre-Emmanuel; Hume, David

Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2613-2640.

Nous fournissons des conditions suffisantes garantissant qu’un groupe donné opérant par isométries sur un espace métrique géodésique soit à hyperbolicité acylindrique. Diverses applications aux groupes d’isométries d’espaces CAT(0) sont mentionnées. Nous montrons en outre qu’un groupe d’automorphismes d’un immeuble irréductible non-sphérique et non-affine est à hyperbolicité acylindrique s’il existe une chambre à stabilisateur fini dont l’orbite contienne un appartement. Ce critère est finalement appliqué aux formes orthogonales des groupes de Kac–Moody sur des corps arbitraires. Il s’applique également aux produits graphés irréductibles de groupes arbitraires, ce qui fournit une nouvelle démonstration d’un résultat récent de Minasyan–Osin.

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.

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