Geodesic
Groups in which every Finitely Generated Subgroup is almost a Free Factor
Canadian journal of mathematics, Tome 31 (1979) no. 6, pp. 1329-1338

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In [5] M. Hall Jr. proved, without stating it explicitly, that every finitely generated subgroup of a free group is a free factor of a subgroup of finite index. This result was made explicit, and used to give simpler proofs of known results, in [1] and [7]. The standard generalization to free products was given in [2]: If, following [13], we call a group in which every finitely generated subgroup is a free factor of a subgroup of finite index an M. Hall group, then a free product of M. Hall groups is again an M. Hall group. The recent appearance of [13], in which this result is reproved, and the rather restrictive nature of the property of being an M. Hall group, led us to attempt to determine the structure of such groups. In this paper we go a considerable way towards achieving this for those M. Hall groups which are both finitely generated and accessible. 
Brunner, A. M.; Burns, R. G. Groups in which every Finitely Generated Subgroup is almost a Free Factor. Canadian journal of mathematics, Tome 31 (1979) no. 6, pp. 1329-1338. doi: 10.4153/CJM-1979-110-2
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