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.

16 pages, 11 figures, to be published in Algebraic & Geometric Topology

References in corpus (2)

• Proof of the $K(\pi ,1)$ conjecture for affine Artin groups
• On the acylindrical hyperbolicity of the tame automorphism group of ${\mathrm{SL}}_2(\mathbb{C})$