Property (LR) for virtually free groups

A classical theorem of M. Hall states that every finitely generated subgroup of a free group is a free factor of a finite index subgroup. Moreover, the works of Brunner--Burns and Dunwoody, imply that any finitely presented group satisfying this M. Hall's property must be virtually free, i.e., it must have a finite index free subgroup. However, many virtually free groups fail to have M. Hall's property. In the talk I will discuss the result that a weaker property (LR), defined by Long and Reid, does holds in all virtually free groups. Namely, every finitely generated subgroup of a virtually free group is a retract of a finite index subgroup. I will sketch the proof of this result, which combines a new embedding theorem for countable virtually free groups with a geometric argument using actions on trees.