Quasi-isometry classification of RAAGs that split over cyclic subgroups

For a one-ended right-angled Artin group, we give an explicit description of its JSJ tree of cylinders over infinite cyclic subgroups in terms of its defining graph. This is then used to classify certain right-angled Artin groups up to quasi-isometry. In particular, we show that if two right-angled Artin groups are quasi-isometric, then their JSJ trees of cylinders are weakly equivalent. Although the converse to this is not generally true, we define quasi-isometry invariants known as stretch factors that can distinguish quasi-isometry classes of RAAGs with weakly equivalence JSJ trees of cylinders. We then show that for many right-angled Artin groups, being weakly equivalent and having matching stretch factors is a complete quasi-isometry invariant.Comment: 49 pages, 12 figures. The class of dovetail RAAGs is introduced and the main theorem is reformulated in terms of such RAAGs. Accepted by Groups, Geometry, and Dynamic

Quasi-isometry Invariants from Decorated Trees of Cylinders of Two-Ended JSJ Decompositions

We construct quasi-isometry invariants of a one-ended finitely presented group by considering the tree of cylinders of a two-ended JSJ decomposition of the group. When the group satisfies additional quasi-isometric rigidity hypotheses we construct finer invariants by also considering relative amounts of stretching across edges of the tree of cylinders.Comment: arXiv:1601.07147 now contains all of these results and much, much mor

Quasi-isometries Between Tubular Groups

We give a method of constructing maps between tubular groups inductively according to a set of strategies. This map will be a quasi-isometry exactly when the set of strategies is consistent. Conversely, if there exists a quasi-isometry between tubular groups, then there is a consistent set of strategies for them. There is an algorithm that will in finite time either produce a consistent set of strategies or decide that such a set does not exist. Consequently, this algorithm decides whether or not the groups are quasi-isometric.Comment: 44 pages, 11 figures. PDFLaTeX. Improved exposition and added some auxiliary material to make the paper more self contained, per referee's comment