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This is a Quantitative Study study.

October 10, 2025Open Access

HNN extensions of free groups with equal associated subgroups of finite index: polynomial time word problem

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The word problem in HNN extensions of free groups with equal associated subgroups is decidable in polynomial time.
This finding applies to cases where the associated subgroups are not normal under specific conditions on the isomorphism.
The research extends existing knowledge on the complexity of word problems in abstract algebra and group theory.
Understanding the polynomial time decidability has implications for computational group theory, enhancing practical applications.

Implication

This research demonstrates that the word problem is decidable in polynomial time for HNN extensions, indicating its significance in group theory.

Abstract

Let G=F_φt be an HNN extension of a free group F with two equal associated normal subgroups H₁ = H₂ of finite index. We prove that the word problem in G is decidable in polynomial time. This result extends to the case where the subgroups H₁=H₂ are not normal, provided that the isomorphism φ:H₁→ H₂ satisfies an additional condition described in Section 5.

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Cite This Study

Shen et al. (2025) studied this question.

synapsesocial.com/papers/68e865117ef2f04ca37e4e15 https://doi.org/10.48550/arxiv.2510.03801

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Implication

This research demonstrates that the word problem is decidable in polynomial time for HNN extensions, indicating its significance in group theory.

HNN extensions of free groups with equal associated subgroups of finite index: polynomial time word problem

Key Points

Implication

Abstract

Discussion

Cite This Study

HNN extensions of free groups with equal associated subgroups of finite index: polynomial time word problem

Key Points

Implication

Abstract

Discussion

Cite This Study

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