JSJ splittings for all Artin groups

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.

arXiv commentV4: Added examples of Artin groups we can distinguish with our methods. Improved exposition and changed numbering. Accepted in Publ. Mat. V2: We deduce the R-infinity property for Artin groups with separating, non-central vertices, as well as the existence of an infinite dimensional space of automorphism-invariant quasimorphisms. 26 pages, 5 figures. Comments are welcome!