# Acylindrical actions for two-dimensional Artin groups of hyperbolic type

## TL;DR

This paper proves that certain two-dimensional hyperbolic Artin groups act acylindrically on hyperbolic spaces, leading to new insights into their subgroup structure and hyperbolic properties.

## Contribution

It establishes acylindrical actions for two-dimensional hyperbolic Artin groups and derives consequences like the Tits alternative and subgroup classifications.

## Key findings

- Artin groups act acylindrically on hyperbolic spaces
- Irreducible Artin groups with ≥3 generators are acylindrically hyperbolic
- Subgroups virtually splitting as direct products are classified

## Abstract

For a two-dimensional Artin group $A$ whose associated Coxeter group is hyperbolic, we prove that the action of $A$ on the hyperbolic space obtained by coning off certain subcomplexes of its modified Deligne complex is acylindrical. Moreover, if for each $s\in S$ there is $t\in S$ with $m_{st}< \infty$, then this action is universal. As a consequence, for $|S|\geq 3$, if $A$ is irreducible, then it is acylindrically hyperbolic. We also obtain the Tits alternative for $A$, and we classify the subgroups of $A$ that virtually split as a direct product. A key ingredient in our approach is a simple criterion to show the acylindricity of an action on a two-dimensional $\mathrm{CAT}(-1)$ complex.