Parabolic subgroup intersections in Artin groups
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In every finite-rank Artin group, is the intersection of two parabolic subgroups again parabolic? Here parabolic subgroups are conjugates of standard parabolic subgroups, and the question includes nonspherical types and infinite Coxeter labels.
Why it matters and what counts as progress
Why it matters: Closure under intersection would give a uniform structural property of parabolic subgroups beyond the Artin-group classes already understood.
What counts as progress: Extend intersection closure beyond known Artin-group classes, or construct a pair with nonparabolic intersection.
How the problem developed
- 2023 · Partial result [1]
P. Möller, L. Paris and O. Varghese Established intersection results for complete parabolics under an intersection hypothesis, leaving the unrestricted Artin-group conjecture open.
- 23 Sept 2026 · Proof claim [2]
OpenAI Released a manuscript claiming that intersections of conjugates of standard parabolics are parabolic for every finite-rank Artin group, including nonspherical types and infinite Coxeter labels.
AI contribution
Parabolic intersections in Artin groups
The September 23 manuscript claims that the intersection of any two conjugates of standard parabolic subgroups is parabolic in every finite-rank Artin group, including nonspherical types and infinite Coxeter labels.
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