One-ended subgroups of graphs of free groups

A longstanding question in geometric group theory is the following. Suppose G is a hyperbolic group where all finitely generated subgroups of infinite index are free. Is G the fundamental group of a surface? This question is still open for some otherwise well understood classes of groups. In this talk, I will explain why the answer is affirmative for graphs of free groups with cyclic edge groups. I will also discuss the extent to which these techniques help with the harder problem of finding surface subgroups.