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

TL;DR

This paper introduces a new notion of extra-large relative to parabolic subgroups in Artin groups, proving the $K()$ conjecture under certain conditions and establishing acylindrical hyperbolicity.

Contribution

It generalizes the concept of extra-large relative to two parabolic subgroups and proves the $K()$ conjecture and acylindrical hyperbolicity for a broader class of Artin groups.

Findings

$K()$ conjecture holds when distinguished subgroups do.

Artin groups are acylindrically hyperbolic under mild conditions.

Generalization of extra-large relative notion for parabolic subgroups.

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.