Ascending Chains in 3-Manifold and Relatively Hyperbolic Groups

TL;DR

The paper proves that ascending chains of bounded rank subgroups in 3-manifold groups stabilize, using geometrization and properties of hyperbolic groups, with new results on relatively hyperbolic groups.

Contribution

It establishes stabilization of ascending chains in 3-manifold groups and introduces a new theorem for relatively hyperbolic groups with bounded rank subgroups.

Findings

Ascending chains of bounded rank subgroups in 3-manifold groups stabilize.

Theorem is new even for hyperbolic groups with bounded rank, locally quasiconvex subgroups.

Uses geometrization to reduce the problem to hyperbolic 3-manifolds.

Abstract

We prove that any ascending chain of bounded rank subgroups in the fundamental group of a compact $3$-manifold stabilizes. We use geometrization to reduce the proof to fundamental groups of complete, finite-volume hyperbolic $3$-manifolds. To handle this case, we prove the following: In a toral relatively hyperbolic group, any ascending chain of bounded rank, locally relatively quasiconvex subgroups stabilizes. We note this theorem is new even for bounded rank, locally quasiconvex chains in hyperbolic groups.