Abstract
In this thesis, we construct suitable collections of transverse parabolic subgroups which extend the maximal simplices of the complex of irreducible parabolic subgroups to analogues of clean markings. We show that there is a marking graph for finite-type Artin groups which is quasi-isometric to the group modulo its center, i.e., an element of $A_{\Gamma}/Z(A_{\Gamma})$ is coarsely determined by its action on one of our markings.
In the mapping class group, markings were a key component in Masur and Minsky's hierarchy machinery, which proved to be a powerful tool in the study of mapping class groups. The hierarchy machinery for mapping class groups was generalized to other classes of groups with the notion of hierarchical hyperbolicity, introduced by Behrstock, Hagen, and Sisto, and then expressed as a simpler set of sufficient conditions known as combinatorial hierarchical hyperbolicity by Behrstock, Hagen, Martin, and Sisto. Two classes of Artin groups, RAAGs and extra-large type Artin groups, are currently known to be hierarchically hyperbolic. Since hierarchical hyperbolicity implies a number of interesting group-theoretic properties, it is natural to ask which classes of Artin groups are hierarchically hyperbolic. Proving that the finite-type Artin groups are hierarchically hyperbolic is an obstruction to proving that classes of Artin groups with finite-type parabolic subgroups are hierarchically hyperbolic. We describe the relationship between a combinatorially hierarchically hyperbolic structure for mapping class groups and the marking graph, and we conjecture that our marking graph for finite-type Artin groups can be extended to a CHHG structure in a similar way.