Ascending chains of free subgroups in closed hyperbolic and graph 3-manifold groups

Takahasi and Higman independently proved that any ascending chain of subgroups of constant rank in a free group must stabilize. Kapovich and Myasnikov gave a proof of this fact in the language of graphs and Stallings folds. Using profinite techniques, Shusterman extended this constant-rank ascending chain condition to limit groups, which include closed surface groups. Motivated by Kapovich and Myasnikov's proof we provide two new proofs of this ascending chain condition for closed surface groups, and establish the ascending chain condition for free subgroups of constant rank in fundamental groups of closed hyperbolic and graph 3-manifolds. These results are now subsumed by the more general framework established in joint work with Heikamp, Kohav, and Munro which both corrected a mistake in a previous version of this paper and generalized it. The proof for closed hyperbolic 3-manifolds and graph manifolds is preserved in this unpublished note for its direct, geometric approach, which remains valid and particularly transparent.