A marking graph for finite-type Artin groups

TL;DR

This paper constructs a marking graph for finite-type Artin groups, analogous to the hierarchy machinery in mapping class groups, enabling group elements to be characterized by their action on markings.

Contribution

It introduces a new marking graph for finite-type Artin groups, extending the concept of clean markings to this algebraic setting.

Findings

The marking graph is quasi-isometric to the group modulo its center.

The construction involves transverse parabolic subgroups and elementary moves.

Provides a new geometric tool for studying 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.