21.49 (2026)

An isometric action of a group $G$ on a metric space $S$ is called acylindrical if for every $\varepsilon > 0$ there exist $R, N > 0$ such that for every two points $x, y$ with $d(x, y) \geqslant R$, there are at most $N$ elements $g \in G$ satisfying $d(x, gx) \leqslant \varepsilon$ and $d(y, gy) \leqslant \varepsilon$. A group is said to be acylindrically hyperbolic if it is not virtually cyclic and admits an acylindrical action on a hyperbolic space with unbounded orbits. Is the automorphism group of a finitely generated acylindrically hyperbolic group also acylindrically hyperbolic?