Partially CAT(-1) groups are acylindrically hyperbolic

Genevois, Anthony; Stocker, Arnaud

Bulletin de la Société Mathématique de France, Tome 147 (2019) no. 3, pp. 377-394

In this paper, we show that, if a group G acts geometrically on a geodesically complete CAT(0) space X which contains at least one point with a CAT(-1) neighborhood, then G must be either virtually cyclic or acylindrically hyperbolic. As a consequence, the fundamental group of a compact Riemannian manifold whose sectional curvature is non positive everywhere and negative in at least one point is either virtually cyclic or acylindrically hyperbolic. This statement provides a precise interpretation of an idea expressed by Gromov in his paper Asymptotic invariants of infinite groups.

Dans cet article, nous démontrons que, si un groupe G agit géométriquement sur un espace CAT(0) X qui est géodésiquement complet et qui contient au moins un point admettant un voisinage CAT(-1), alors G doit être ou bien acylindriquement hyperbolique ou bien virtuellement cyclique. Par conséquent, le groupe fondamental d’une variété riemannienne compacte dont la courbure sectionnelle est négative ou nulle partout et strictement négative en au moins un point doit être acylindriquement hyperbolique ou virtuellement cyclique. Cet énoncé propose une interprétation précise et moderne d’une idée de Gromov décrite dans Asymptotic invariants of infinite groups.

Classification : 20F65, 20F67
Keywords: acylindrically hyperbolic groups, CAT(0) groups, rank-one isometries
Mots-clés : groupes acylindriquement hyperboliques, groupes CAT(0), isométries de rang un

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