  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 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 a closed (or finite-volume
hyperbolic) 3-manifold group. Hyperbolic geometry, geometrization, and
graphs-of-groups decompositions all play a role in our proofs.
