Description
Two groups are called commensurable if they have isomorphic subgroups of finite index. This gives an equivalence relation which is in general finer than the more geometric equivalence relation of being quasi-isometric. Given a simple graph, a right-angled Artin group (RAAG) can be defined by taking one generator for each vertex and one defining relation for each edge saying that the corresponding two generators commute. RAAGs form an important class of groups, which recently played a key role in the work of Daniel Wise on special cube complexes and subsequent solution by Ian Agol of the virtually fibered conjecture, a well-known problem in 3-manifold topology formulated by Thurston.
The general question of when two RAAGs are commensurable or quasi-isometric appears to be hard, and only some partial results are known so far. In 2008 J. Behrstock and W. Neumann proved that all RAAGs defined by trees of diameter at least 3 are quasi-isometric. We describe the commensurabilty classes of RAAGs defined by trees of diameter 4 and by path graphs, confirming the conjecture of J. Behrstock and W. Neumann that there are infinitely many such commensurability classes. This is joint work with Montse Casals-Ruiz and Ilya Kazachkov.