Abstract

We consider some questions concerning finiteness properties in infinite groups which are related to Marshall Hall's theorem. We call these properties Property S and Property R, and they are trivially true in finite groups. We give several elementary proofs using these properties for results on finitely generated subgroups of free groups as well as a new elementary proof of Hall's basic result. Finally, we consider these properties within surface groups and prove an analog of Hall's theorem in that context.

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