The Parabolic Intersection Conjecture for Artin groups

Theorem 1.1: for every Coxeter matrix on a finite set, all $X,Y$ and $g,h$, there are $Z,k$ with $gA_Xg^{-1}\cap hA_Yh^{-1}=kA_Zk^{-1}$. Corollary 1.2 (via Moller-Paris-Varghese): arbitrary intersections of parabolics are parabolic, already attained by at most $|S|$ members, and every subset has a unique parabolic closure. Corollary 1.4: irreducible Artin groups with an infinite label are acylindrically hyperbolic (already known from Kato-Oguni) with weakly malnormal proper parabolics. The proof uses a relative harmonic height for a framed zigzag braid action and does not use asphericity of the Salvetti complex.

Theorem 1.1: for every Coxeter matrix on a finite set, all $X,Y$ and $g,h$, there are $Z,k$ with $gA_Xg^{-1}\cap hA_Yh^{-1}=kA_Zk^{-1}$. Corollary 1.2 (via Moller-Paris-Varghese): arbitrary intersections of parabolics are parabolic, already attained by at most $|S|$ members, and every subset has a unique parabolic closure. Corollary 1.4: irreducible Artin groups with an infinite label are acylindrically hyperbolic (already known from Kato-Oguni) with weakly malnormal proper parabolics. The proof uses a relative harmonic height for a framed zigzag braid action and does not use asphericity of the Salvetti complex.