CAT(0) cube complexes are a well-studied and well-behaved class of spaces. We can get special examples of such spaces by looking at right angled Artin groups (RAAGs), which are a hybrid between free abelian and free groups – all the generators have infinite order, and the relators are some commutators of generators. Many RAAGs are known to be quasi-isometrically rigid. In this talk, I will discuss rigidity of quasi-isometric _embeddings_ between RAAGs of the same rank, and its limitations.
This talk is based on joint work with Oussama Bensaid and Harry Petyt.