We give a technical result that implies a straightforward necessary and sufficient
condition for a graph of groups with virtually cyclic edge groups to be one-ended. For
arbitrary graphs of groups, we show that if their fundamental group is not one-ended,
then we can blow up vertex groups to graphs of groups with simpler vertex and edge
groups. As an application, we generalize a theorem of Swarup to decompositions of
virtually free groups.
Keywords
group theory, groups acting on trees, Bass–Serre theory,
ends of groups, square complexes