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Acylindrical actions for two-dimensional Artin groups of hyperbolic type

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.