Cubical-like geometry of graph products

In 1994, Bandelt, Mulder and Wilkeit introduced a class of graphs generalizing the so-called median graphs: the class of quasi-median graphs. Since the works of Roller and Chepoï, we know that median graphs and CAT(0) cube complexes essentially define the same objets, and because CAT(0) cube complexes play an important role in recent reseach in geometric group theory, a natural question is whether quasi-median graphs can be used to study some classes of groups. In our talk, we will show that quasi-median graphs and CAT(0) cube complexes share essentially the same geometry. Moreover, extending the observation that right-angled Artin and Coxeter groups have a Cayley graph which is median, arbitrary graph products turn out to have a Cayley graph (with respect to a natural, but possibly infinite, generating set) which is quasi-median. The main goal of this talk is to show how to use the quasi-median geometry of this Cayley graph to study graph products.