A marking graph for finite-type Artin groups

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 $A_Γ/Z(A_Γ)$ 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.

Publication Details

Journal
ArXiv.org
Published
2025-08-14
Primary Topic
Geometric and Algebraic Topology
Type
article
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article

A marking graph for finite-type Artin groups

ArXiv.org
Geometric and Algebraic Topology
article

A marking graph for finite-type Artin groups

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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 $A_Γ/Z(A_Γ)$ 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.

ArXiv.org
Openalex Percentile: Top 42%
Geometric and Algebraic Topology
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A marking graph for finite-type Artin groups · ArXiv.org (2025) | TGRS Research Map | TGRS