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

Let [Formula: see text] be a finitely generated group, [Formula: see text] a finite set of generators and [Formula: see text] a subgroup of [Formula: see text]. We define what it means for [Formula: see text] to be a context-free pair; when [Formula: see text] is trivial, this specializes to the standard definition of [Formula: see text] 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 [Formula: see text] and finite index enlargements of [Formula: see text]. If [Formula: see text] is virtually free and [Formula: see text] is finitely generated then [Formula: see text] is context-free. A basic tool is the following: [Formula: see text] is context-free if and only if the Schreier graph of [Formula: see text] with respect to [Formula: see text] is a context-free graph.