21.128 (2026)

Two groups $G_1$ and $G_2$ are said to be commensurable if there exist finite index subgroups $H_1 \leqslant G_1$ and $H_2 \leqslant G_2$ (not necessarily of the same index) such that $H_1 \cong H_2$. Let $A[F_4]$ and $A[H_4]$ denote the Artin groups of spherical types $F_4$ and $H_4$, respectively. Are these two groups commensurable? This is the most difficult case in the classification of Artin groups of spherical type up to commensurability.