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7. Parabolic subgroups


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      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


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          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, $\mathrm{CAT}(0)$)


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              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.