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3. The isomorphism problem

    1. Problem 3.1.

      Is the class of spherical Artin groups closed under isomorphism? In other words, if an Artin group A is isomorphic to a spherical Artin group, must A be spherical?
        • Problem 3.2.

          Solve the isomorphism problem for 2-dimensional Artin groups.
              Open even for Coxeter groups
            • Problem 3.3.

              Solve the isomorphism problem for \infty-free 2-dimensional Artin groups.
                  Known for Coxeter groups
                • Problem 3.4.

                  Find the commensurability and Q.I. classification for \infty-free Artin groups.

                  A specific case: are F_4 and H_4 commensurable?
                    • Problem 3.5.

                      For an Artin group A, when is \mathrm{Out}(A) finite/finitely generated? Find explicit generators.
                        • Problem 3.6.

                          For an Artin group A, which Artin groups A' embed in A such that A' is not a parabolic subgroup?
                            • Problem 3.7.

                              For fixed A, which spherical and/or dihedral Artin groups embed in A?
                                • Problem 3.8.

                                  Classify \mathrm{End}(A).
                                      Known for A_n and D_n types. Likely doable for B_n, \tilde{A}_n, \tilde{C}_n

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