Ascending chains of free groups in 3-manifold groups

TL;DR

This paper proves that ascending chains of free subgroups of constant rank in 3-manifold groups, including hyperbolic ones, must stabilize, extending known results from free and surface groups using geometric and graph-theoretic methods.

Contribution

It provides two new proofs for the ascending chain condition in closed surface groups and establishes this property for free subgroups in 3-manifold groups, combining hyperbolic geometry and graphs-of-groups.

Findings

Ascending chains of free subgroups in 3-manifold groups stabilize.

New proofs for surface groups' ascending chain condition.

Extension of chain stabilization to hyperbolic 3-manifold groups.

Abstract

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.