Ends of Group Pairs and Non-Positively Curved Cube Complexes

We explore a generalized Bass-Serre theory relating the notion of multi-endedness for pairs of groups to actions of groups on non-positively curved cube complexes (called cubings). We show that a group is multi-ended with respect to a subgroup if and only if it acts ‘essentially’ on a cubing. We explore several examples of this phenomenon. We discuss actions on finite-dimensional cubings, and show that in this case an essential action is simply an unbounded action. We discuss an application to the study of immersed incompressible surfaces in 3-manifolds.