[Paper Review] On the peripheral subgroups of irreducible 3-manifold groups and acylindrical splittings

Filippo Cerocchi|arXiv (Cornell University)|May 17, 2017

Geometric and Algebraic Topology19 references3 citations

TL;DR

This paper investigates the conjugacy and malnormality properties of abelian peripheral subgroups in the fundamental group of orientable, irreducible 3-manifolds, focusing on their behavior within Seifert fibered and hyperbolic JSJ-components. Using Bass-Serre theory and acylindrical splittings, it provides new proofs of conjugacy separation results, showing that intersections of conjugated peripheral subgroups are trivial or generated by regular fibers, depending on whether the JSJ-component is hyperbolic or Seifert fibered.

ABSTRACT

We survey the problem of separation under conjugacy and malnormality of the abelian peripheral subgroups of an orientable, irreducible $3$-manifold $X$. We shall focus on the relation between this problem and the existence of acylindrical splittings of $π_1(X)$ as an amalgamated product or HNN-extension along the abelian subgroups corresponding to the JSJ-tori.

Motivation & Objective

• To understand the conjugacy behavior of abelian peripheral subgroups in the fundamental group of orientable, irreducible 3-manifolds.
• To clarify when these peripheral subgroups are malnormal, particularly in relation to JSJ-components of hyperbolic or Seifert fibered type.
• To reprove and reframe results from [dlHW14b] and [WZ10] using Bass-Serre theory and acylindrical group actions.
• To characterize the intersection of conjugated peripheral subgroups in terms of regular fibers of Seifert fibered JSJ-components.
• To establish conditions under which conjugated peripheral subgroups intersect nontrivially, linking this to the structure of the JSJ decomposition.

Proposed method

• Analyzes the fundamental group of a 3-manifold via its JSJ decomposition into atoroidal and Seifert fibered components.
• Applies Bass-Serre theory to model the fundamental group as a graph of groups with edge groups corresponding to JSJ-tori.
• Uses acylindrical splittings to study the action of π₁(X) on the associated Bass-Serre tree and the stabilizers of edges and vertices.
• Applies the theory of malnormal subgroups and conjugacy separability to analyze intersections of conjugated peripheral subgroups.
• Employs reduced word forms in the graph of groups to determine when conjugated elements lie in peripheral subgroups.
• Leverages the uniqueness of regular fiber elements in Seifert fibered components to define canonical generators for intersections.

Experimental results

Research questions

• RQ1 Under what conditions is a peripheral subgroup corresponding to a boundary torus malnormal in the fundamental group of a 3-manifold?
• RQ2 When do two conjugated peripheral subgroups intersect nontrivially, and what generates their intersection?
• RQ3 How does the JSJ decomposition influence the conjugacy properties of peripheral subgroups?
• RQ4 What role does the Seifert fibration structure play in determining the intersection of conjugated peripheral subgroups?
• RQ5 Can the results of [dlHW14b] and [WZ10] be reproven using Bass-Serre theory and acylindrical group actions?

Key findings

• A peripheral subgroup π₁(T) corresponding to a boundary torus T is malnormal in π₁(X) if and only if T bounds a JSJ-component of hyperbolic type.
• If two boundary tori lie in the same Seifert fibered JSJ-component, then gπ₁(T₁)g⁻¹ ∩ π₁(T₂) ≠ {1} if and only if g lies in the fundamental group of that component and the intersection is generated by the regular fiber fₖ.
• For distinct Seifert fibered components, the intersection of conjugated peripheral subgroups is trivial unless both tori lie in the same component, in which case the intersection is again generated by the common regular fiber.
• The regular fiber fₖ is uniquely determined up to inverse within each Seifert fibered JSJ-component, providing a canonical generator for intersections.
• Conjugated elements gwg⁻¹ lie in a peripheral subgroup π₁(T) only if the word form is not reduced or specific conditions on the JSJ-component and fiber element are met, otherwise |gwg⁻1| ≥ 2 and gwg⁻¹ ∉ π₁(T).
• The paper establishes that π₁(T⁺¹) and π₁(T⁻¹) are conjugately separated unless both tori bound the same Seifert fibered component, in which case the intersection is controlled by the regular fiber.