Context-free pairs of groups I: Context-free pairs and graphs

Let G be a finitely generated group, A a finite set of generators and K a subgroup of G. We define what it means for (G,K) to be a context-free pair; when K is trivial, this specializes to the standard definition of G to be a context-free group.We derive some basic properties of such group pairs. Context-freeness is independent of the choice of the generating set. It is preserved under finite index modifications of G and finite index enlargements of K. If G is virtually free and K is finitely generated then (G,K) is context-free. A basic tool is the following: (G,K) is context-free if and only if the Schreier graph of (G,K) with respect to A is a context-free graph.