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7. Parabolic subgroups

    1. Problem 7.1.

      Are intersections of parabolic subgroups themselves parabolic? More specifically: is this true in the Euclidean case?
          Known when the Artin group is FC-type and one of the parabolics is spherical
        • Problem 7.2.

          Are intersections of parabolic again parabolic when the Deligne complex has some notion of non-positive curvature? (i.e., systolic, injective, CUB, \mathrm{CAT}(0))
            • Problem 7.3.

              Let A be an Artin group with subgroup H. If for all h \in H there is a (spherical) parabolic containing h, does there exist a (spherical) parabolic containing H? Geometric version: If a group G acts on X and H < G with each h \in H fixing a point of X, then does H fix a point? What about when X has non-positive curvature?

                  Cite this as: AimPL: Geometry and topology of Artin groups, available at http://aimpl.org/geomartingp.