JSJ splittings for all Artin groups

TL;DR

This paper establishes a comprehensive JSJ decomposition for all Artin groups, characterizes their splitting properties, and explores implications for automorphism groups and the R-infinity property.

Contribution

It provides the first explicit construction of JSJ decompositions for all Artin groups and links group isomorphisms to their parabolic subgroups.

Findings

Artin groups split over infinite cyclic subgroups iff their defining graph has a separating vertex.

Isomorphic Artin groups share the same set of parabolics supported on big chunks.

Many Artin groups have automorphism groups that are acylindrically hyperbolic.

Abstract

We prove that an Artin group splits over infinite cyclic subgroups if and only if its defining graph has a separating vertex, and explicitly construct a JSJ decomposition over infinite cyclic subgroups for all Artin groups. We then use these facts to show that, if two Artin groups are isomorphic, then they have the same set of parabolics supported on "big chunks", that is, maximal subgraphs without separating vertices. We also deduce acylindrical hyperbolicity for the automorphism groups of many Artin groups, partially answering a question of Genevois in the case of Artin groups. As a consequence, we produce new families of Artin groups with the R-infinity property.