In 2007, Bestvina, Kleiner, and Sageev showed that a certain class of 2-dimensional RAAGs is quasi-isometrically rigid among itself, i.e. if any two groups in the class are quasi-isometric, then they are isomorphic. Huang later generalised these results to higher dimensions, proving that for a large class of RAAGs (those with finite outer automorphism group, no squares, and star-rigid defining graphs) any group quasi-isometric to a group in this class is commensurable with it. This stands in contrast to the fact that all RAAGs defined by trees are quasi-isometric to one another. We study quasi-isometric embeddings of RAAGs, a setting in which the methods of BestvinaKleinerSageev and Huang do not apply. We show that the existence of such embeddings between RAAGs of the same dimension can nevertheless yield strong rigidity results, relating their defining graphs. On the other hand, we construct exotic quasi-isometric embeddings between RAAGs that exist even in the absence of any subgroup relation. This talk is based on joint work with Oussama Bensaid and Harry Petyt.
![[1953] Empilements de sphères en très grande dimension](/media/cache/video_light/uploads/video/SeminaireBourbaki.png)
