The parabolic intersection conjecture for Artin groups

Let AA be an Artin group, and let parabolic subgroups mean the parabolic subgroups of AA. The parabolic intersection conjecture. The intersection of any two parabolic subgroups is again a parabolic subgroup. This concerns the closure of the class of parabolic subgroups under intersections, a basic structural question in the study of Artin groups. The supplied text does not specify whether the assertion is known in full generality.

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Primary source

María Cumplido, “Fundamental techniques in the study of parabolic subgroups of Artin groups”, arXiv:2509.08382 (2026).

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