Outer space for right-angled Artin groups

In order to study the group of (outer) automorphisms of any group G by geometric methods, one needs a well-behaved "outer space" with an interesting action of Out(G). If G is free abelian, the classic symmetric space SL(n,R)/SO(n) serves this role, and if G is free non-abelian an appropriate outer space was introduced in the 1980's. I will first go over these basic constructions and indicate how they are used to study Out(G). I will then introduce joint work with Ruth Charney on constructing an outer space for a more general class of groups, called right-angled Artin groups. These are a hybrid of free and free abelian groups, defined by specifying that some (not necessarily all) generators commute.