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 call the pair $(G,K)$ context-free if the set of all words over $A$ that reduce in $G$ to an element of $K$ is a context-free language. When $K$ is trivial, $G$ itself is called context-free; context-free groups have been classified more than 20 years ago in celebrated work of Muller and Schupp as the virtually free groups. Here, 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.