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8. Algebraic properties

    1. Problem 8.05.

      Are all Artin groups virtually residually finite?
        •     We say a group is good in the sense of Serre if it has the same cohomology as its profinite completion.

          Problem 8.1.

          Are all Artin groups good in the sense of Serre?
            • Problem 8.15.

              Which Artin groups are profinitely rigid?
                • Problem 8.2.

                  Which Artin groups are Hopfian?
                    • Problem 8.25.

                      Does the group ring of an Artin group have zero divisors?
                          Open even for spherical Artin groups
                        • Conjecture 8.3.

                          The generalized Tits conjecture
                            • Problem 8.35.

                              Does the Tits alternative hold for all Artin groups?
                                • Problem 8.4.

                                  Given a,b \in A, is there an n = n(a,b) such that \langle a^n, b^n \rangle is free or free abelian?
                                      Known for RAAGs with n(a,b) = 1, extra-large type, 2-dimensional hyperbolic, and A_n type. Open for spherical in general
                                    • Problem 8.45.

                                      Compute the centralizer of an arbitrary element.

                                      A more specific question: in a Euclidean Artin group, if two elements are contained in centers of spherical parabolic subgroups, when do the two elements commute?
                                        • Problem 8.5.

                                          Find a “good”/“natural” (e.g., finite dimensional) linear representation for Artin groups.
                                            • Problem 8.55.

                                              Prove Coxeter groups are residually finite without appealing to linearity.
                                                • Problem 8.6.

                                                  Which Artin groups have property R_\infty?
                                                    1. Remark. [Ignat Soroko] Known to have property R_\infty: Spherical A_n, B_n, for n\ge2; D_4, I_2(m) for m\ge3, and the pure subgroups of all these; Euclidean: \tilde A_n, \tilde C_n; Certain classes of RAAGs (conjecturally all nonabelian RAAGs [Dekimpe–Senden]).

                                                      Known not to have property R_\infty: abelian RAAGs

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