Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ascending Chains in 3-Manifold and Relatively Hyperbolic Groups

Edgar A. Bering, Jakob Heikamp, Jack Kohav, Nir Lazarovich, Zachary Munro

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.

Ascending Chains in 3-Manifold and Relatively Hyperbolic Groups

Abstract

We prove that any ascending chain of bounded rank subgroups in the fundamental group of a compact -manifold stabilizes. We use geometrization to reduce the proof to fundamental groups of complete, finite-volume hyperbolic -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.

Paper Structure

This paper contains 12 sections, 24 theorems, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph of groups
  4. Relatively hyperbolic groups
  5. Relatively quasiconvex subgroups
  6. A User's Guide to Weidmann-Weller Carrier Graphs
  7. Ascending chains in toral relatively hyperbolic groups
  8. Rank bounds for acylindrical graphs of groups
  9. Complexity of subgroups
  10. Freely indecomposable factors
  11. Chains of isomorphic subgroups
  12. Three manifold groups

Key Result

Theorem A

Fundamental groups of compact 3-manifolds satisfy $\omega$ACC.

Figures (1)

  • Figure 1: Peripheral stars have central vertices marked by $\circ$ and gray edges. The base vertex $v_0$ is marked by $\square$. The edge labels and vertex groups are marked on the graph. Edge groups of thick edges are all $\left< c \right>$, and the thin edge is free.

Theorems & Definitions (80)

  • Definition 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1: Graphs
  • Definition 2.2: Paths
  • Definition 2.3: Graph of groups
  • Definition 2.4: Orientation
  • Definition 2.5: $\mathbb{A}$-path
  • Definition 2.6: Fundamental group $\pi_1(\mathbb{A}, v_0)$
  • ...and 70 more