Acylindrical hyperbolicity of outer automorphism groups of right-angled Artin groups

TL;DR

This paper characterizes when the outer automorphism group of a right-angled Artin group is acylindrically hyperbolic, based on the graph's SIL-pair structure, with probabilistic and classification results.

Contribution

It provides a necessary and sufficient condition for acylindrical hyperbolicity of Out(A_Γ) based on SIL-pairs and classifies partial conjugations when SIL-pairs are maximal.

Findings

Out(A_Γ) is acylindrically hyperbolic if and only if Γ has no SIL-pair.

Random connected graphs satisfying certain conditions yield non-acylindrically hyperbolic Out(A_Γ) with high probability.

Classification of partial conjugations when Γ has a maximal SIL-pair system.

Abstract

We study the acylindrical hyperbolicity of the outer automorphism group of a right-angled Artin group $A_\Gamma$. When the defining graph $\Gamma$ has no SIL-pair (separating intersection of links), we obtain a necessary and sufficient condition for $\mathrm{Out}(A_\Gamma)$ to be acylindrically hyperbolic. As a corollary, if $\Gamma$ is a random connected graph satisfying a certain probabilistic condition, then $\mathrm{Out}(A_\Gamma)$ is not acylindrically hyperbolic with high probability. When $\Gamma$ has a maximal SIL-pair system, we derive a classification theorem for partial conjugations. Such a classification theorem allows us to show that the acylindrical hyperbolicity of $\mathrm{Out}(A_\Gamma)$ is closely related to the existence of a specific type of partial conjugations.