[Paper Review] Mapping tori of free group automorphisms are coherent

Mark Feighn, Michael Handel|arXiv (Cornell University)|May 1, 1999

Geometric and Algebraic Topology1 references4 citations

TL;DR

This paper proves that mapping tori of injective endomorphisms of free groups are coherent, meaning all finitely generated subgroups are finitely presented and of finite type. Using techniques from geometric group theory and Bass-Serre theory, the authors establish that such HNN extensions over free groups preserve coherence under injective endomorphisms, resolving a key structural question in group theory.

ABSTRACT

The mapping torus of an endomorphism Φof a group G is the HNN-extension G*_G with bonding maps the identity and Φ. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated subgroups are finitely presented and, moreover, these subgroups are of finite type.

Motivation & Objective

• To investigate the coherence properties of mapping tori constructed from injective endomorphisms of free groups.
• To determine whether all finitely generated subgroups of such mapping tori are finitely presented.
• To establish that these subgroups are of finite type, a stronger finiteness condition in group theory.
• To extend structural understanding of HNN extensions over free groups under injective endomorphisms.
• To resolve a long-standing question about coherence in the class of mapping tori of free group automorphisms.

Proposed method

• Construct the mapping torus as an HNN extension of a free group with bonding maps given by the identity and an injective endomorphism.
• Apply Bass-Serre theory to analyze the structure of the resulting HNN extension.
• Use the fact that injective endomorphisms of free groups preserve certain finiteness properties under HNN extension.
• Leverage results on finitely generated subgroups in HNN extensions to prove finite presentability.
• Employ geometric and combinatorial group theory techniques to verify that subgroups are of finite type.
• Utilize the structure of the mapping torus as a group extension to deduce coherence.

Experimental results

Research questions

• RQ1 Are all finitely generated subgroups of the mapping torus of an injective free group endomorphism finitely presented?
• RQ2 Does the mapping torus of an injective endomorphism of a free group satisfy the property of coherence?
• RQ3 Can the coherence of such mapping tori be established using HNN extension theory and Bass-Serre theory?
• RQ4 What structural properties of injective endomorphisms of free groups ensure coherence in their mapping tori?
• RQ5 Is the class of mapping tori of injective free group endomorphisms closed under finite generation and finite presentability?

Key findings

• The mapping torus of an injective endomorphism of a free group is coherent, meaning all finitely generated subgroups are finitely presented.
• All finitely generated subgroups of such mapping tori are of finite type, a stronger finiteness condition than finite presentability.
• The coherence result holds specifically for injective endomorphisms, not necessarily for arbitrary endomorphisms.
• The proof relies on the structure of HNN extensions and the properties of injective endomorphisms in free groups.
• The result establishes a significant structural constraint on the subgroup lattice of these mapping tori.
• The paper confirms that the mapping torus construction preserves coherence under injective endomorphisms of free groups.