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8. Algebraic properties


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      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.

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          Problem 8.1.

          Are all Artin groups good in the sense of Serre?


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              Problem 8.15.

              Which Artin groups are profinitely rigid?


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                  Problem 8.2.

                  Which Artin groups are Hopfian?


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                      Problem 8.25.

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


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                          Conjecture 8.3.

                          The generalized Tits conjecture


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                              Problem 8.35.

                              Does the Tits alternative hold for all Artin groups?


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                                  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


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                                      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?


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                                          Problem 8.5.

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


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                                              Problem 8.55.

                                              Prove Coxeter groups are residually finite without appealing to linearity.


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                                                  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.