Kourovka Notebook Problem 21.49

Definitions and mathematical statement

A geodesic metric space is hyperbolic if some $\delta\ge0$ makes each side of every geodesic triangle lie in the $\delta$-neighbourhood of the other two sides. An isometric action of a group $K$ on a metric space $X$ is acylindrical if for every $\varepsilon>0$ there exist $R>0$ and an integer $N\ge1$ such that, whenever $d(x,y)\ge R$, \[ |\{g\in K:d(x,gx)\le\varepsilon, \ d(y,gy)\le\varepsilon\}|\le N. \] The constants are uniform over all such pairs of points.

A group is virtually cyclic if it has a cyclic subgroup of finite index. In the source convention, $K$ is acylindrically hyperbolic if it is not virtually cyclic and admits an acylindrical isometric action on a hyperbolic geodesic space with an unbounded orbit. The space need not be proper or locally finite, and faithfulness of the action is not part of the definition.

Let $G$ be a finitely generated acylindrically hyperbolic group. Is its full automorphism group $\operatorname{Aut}(G)$ also acylindrically hyperbolic? This group consists of all bijective homomorphisms $G\to G$, with composition as multiplication.

The requested action for $\operatorname{Aut}(G)$ may use a different hyperbolic space from the one supplied for $G$. Neither finite generation of $\operatorname{Aut}(G)$ nor finite outer automorphism group is assumed. The target is the full automorphism group, not just its subgroup of inner automorphisms or a selected finite-index subgroup.

Short English statement

Is the full automorphism group of every finitely generated acylindrically hyperbolic group itself acylindrically hyperbolic?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2558 (KOU-21.49), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.49. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.