The $K(\pi,1)$ conjecture and acylindrical hyperbolicity for relatively extra-large Artin groups

Abstract

Let $A_\Gamma$ be an Artin group with defining graph $\Gamma$. We introduce the notion of $A_\Gamma$ being extra-large relative to a family of arbitrary parabolic subgroups. This generalizes a related notion of $A_\Gamma$ being extra-large relative to two parabolic subgroups, one of which is always large type. Under this new condition, we show that $A_\Gamma$ satisfies the $K(\pi,1)$ conjecture whenever each of the distinguished subgroups do. In addition, we show that $A_\Gamma$ is acylindrically hyperbolic under only mild conditions.