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

5. Complexes for Artin groups

Let $A$ be an Artin group with Deligne complex $D$, Salvetti complex $S$, “pure” Salvetti complex $P$, and complex of irreducible parabolics $X$.


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      Problem 5.1.

      What kind of non-positive curvature might these complexes have? (systolic, Helly, CUB, $\mathrm{CAT}(0)$?)


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          Problem 5.2.

          $K(\pi,1)$ conjecturette: is $\pi_2$ of Deligne complex/Salvetti complex/hyperplane complement always trivial? Find a proof using geometric/combinatorial techniques.
              Claimed to be known via a category theoretical proof.


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              Problem 5.3.

              Find a direct combinatorial proof (i.e., without reference to mapping class groups) that $X$ is $\delta$-hyperbolic for the braid groups.


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                  Problem 5.4.

                  If one of the complexes has non-positive curvature and all cell stabilizers have solvable word problem, does this mean the whole group has solvable word problem? Can this be generalized to arbitrary cocompact actions on NPC complexes? $\mathrm{CAT}(0)$ cube complexes?
                      In the general case, most likely need additional assumptions, namely the membership problem for cell stabilizers into larger cell stabilizers.


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                      Problem 5.5.

                      Let $\Sigma$ be the Davis complex of the associated Artin group. Take intersections of halfspaces in $\Sigma$, then pull back via the natural surjection from $P$. What can we say about these subspaces of $P$? Specifically, when are they $\pi_1$-injective?


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                          Problem 5.6.

                          Is the Moussong metric on the spherical Deligne complex for the type $B_3$ Artin group $\mathrm{CAT}(1)$?


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                              Problem 5.7.

                              Does the $Br_4$ triangle complex embed in a (locally finite?) $\tilde{A}_2$ building?
                                1. Remark. Related to the Burau representation; if it doesn’t embed, then the Burau rep is not faithful over a finite field


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                                      Problem 5.8.

                                      Is there a “natural” (cellular?) compactification of the hyperplane complement associated to an Artin group?

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