A marking graph for finite-type Artin groups

Abstract

Clean markings on surfaces 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. We construct a marking graph for irreducible 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 determined up to finite error by its action on one of our markings. To construct this graph, 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, and we define natural analogues of elementary moves.