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\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

4. Non-positive curvature

    1. Problem 4.05.

      Are the braid groups \mathrm{CAT}(0)?
        • Problem 4.1.

          Prove Br_4 does not act properly on a \mathrm{CAT}(0) cube complex.
            • Problem 4.15.

              Prove that hyperbolic triangle Artin groups do not act properly on a \mathrm{CAT}(0) cube complex.
                • Problem 4.2.

                  Does the \tilde{A}_2 Artin group act properly and cocompactly on a Helly graph?
                    • Problem 4.25.

                      Further classify the systolic Artin groups.
                          Known for RAAGs, 2-dimensional Artin groups, and (2,4,4) triangle Artin group
                        • Problem 4.3.

                          Which Artin groups are HHGs?
                            • Problem 4.35.

                              For a,b \in A, say a \leq b if there is a positive word c with a = bc. Is (A,\leq) a join-semilattice?
                                • Acylindrical hyperbolicity

                                  Conjecture 4.4.

                                  All non-spherical irreducible Artin groups are acylindrically hyperbolic.
                                      Known for 2-dimensional and many sporadic examples. Open for FC-type.
                                    • Problem 4.45.

                                      Describe the hyperbolic space on which the acylindrically hyperbolic Artin groups act.
                                        • Problem 4.5.

                                          Is there a “largest” acylindrically hyperbolic space on which these Artin groups act that sees all loxodromic elements?

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