Spaces of finite graphs play a key role in perturbative quantum field theory, but also in many other areas of science and mathematics. Among these is geometric group theory, where they are used to model groups of automorphism of free groups. Graphs can be thought of as 1-dimensional flat metric spaces.In higher dimensions, spaces of flat n-dimensional tori model automorphism groups of free abelian groups.There are very interesting groups which interpolate between free groups and free abelian groups, called right-angled Artin groups. I will describe a space of “Gamma-complexes”, which are a hybrid of tori and graphs, and which model automorphism groups of right-angled Artin groups, by recent joint work with Bregman and Charney.