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. ive 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.