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

4. Non-positive curvature


    1. Add remark

      Problem 4.05.

      Are the braid groups $\mathrm{CAT}(0)$?


        • Add remark

          Problem 4.1.

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


            • Add remark

              Problem 4.15.

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


                • Add remark

                  Problem 4.2.

                  Does the $\tilde{A}_2$ Artin group act properly and cocompactly on a Helly graph?


                    • Add remark

                      Problem 4.25.

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


                        • Add remark

                          Problem 4.3.

                          Which Artin groups are HHGs?


                            • Add remark

                              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


                                  Add remark

                                  Conjecture 4.4.

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


                                    • Add remark

                                      Problem 4.45.

                                      Describe the hyperbolic space on which the acylindrically hyperbolic Artin groups act.


                                        • Add remark

                                          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.