Partially CAT(-1) groups are acylindrically hyperbolic

TL;DR

This paper proves that groups acting on certain CAT(0) spaces with negative curvature regions are either virtually cyclic or acylindrically hyperbolic, clarifying a concept by Gromov about group actions and curvature.

Contribution

It establishes a new criterion linking geometric actions on CAT(0) spaces with negative curvature regions to acylindrical hyperbolicity, extending Gromov's ideas.

Findings

Groups acting on CAT(0) spaces with negative curvature points are either virtually cyclic or acylindrically hyperbolic.

Fundamental groups of certain nonpositively curved manifolds are classified as either virtually cyclic or acylindrically hyperbolic.

Provides a geometric interpretation of Gromov's idea on group actions and curvature.

Abstract

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 nonpositive 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.