Abstract: We generalize the notions of asymptotic dimension and coarse embeddings, from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that the quantum asymptotic dimension behaves well with respect to several natural operations, and in particular with respect to quantum coarse embeddings. Moreover, in analogy with the classical case, we prove that a quantum metric space that equi-coarsely contains a sequence of quantum expanders must have infinite asymptotic dimension. This is done by proving a vertex-isoperimetric inequality for quantum expanders, based upon a previously known edge-isoperimetric one. Joint work with Andrew Swift.
