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3. The isomorphism problem


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      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?


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          Problem 3.2.

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


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              Problem 3.3.

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


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                  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?


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                      Problem 3.5.

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


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                          Problem 3.6.

                          For an Artin group $A$, which Artin groups $A'$ embed in $A$ such that $A'$ is not a parabolic subgroup?


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                              Problem 3.7.

                              For fixed $A$, which spherical and/or dihedral Artin groups embed in $A$?


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                                  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.