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

Hanwen Shen, Alexander Ushakov

TL;DR

This work resolves the word problem for a broad class of HNN extensions of free groups by combining subgroup-graph representations with compressed word techniques. The authors construct polynomial-time algorithms for $F\ast_\varphi t$ when the associated subgroups $H_1=H_2$ are equal and of finite index, and extend to the non-normal case under the normalizable $\varphi$ condition by reducing to a normal finite-index subgroup $N\unlhd F$ and exploiting syllables, precomputed data, and SLPs. The core methodology leverages Britton reductions alongside sophisticated data-structures (subgroup graphs, shift operations) and compressed representations (SLPs) to avoid exponential blowups. The results have implications for the complexity of fundamental decision problems in HNN-extensions and demonstrate a viable path to polynomial-time solutions via structural compression and normalizer-based reductions.

Abstract

Let $G=F\ast_\varphi t$ be an HNN extension of a free group $F$ with two equal associated normal subgroups $H_1 = H_2$ 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 $H_1=H_2$ are not normal, provided that the isomorphism $\varphi:H_1\to H_2$ satisfies an additional condition described in Section 5.

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

TL;DR

This work resolves the word problem for a broad class of HNN extensions of free groups by combining subgroup-graph representations with compressed word techniques. The authors construct polynomial-time algorithms for when the associated subgroups are equal and of finite index, and extend to the non-normal case under the normalizable condition by reducing to a normal finite-index subgroup and exploiting syllables, precomputed data, and SLPs. The core methodology leverages Britton reductions alongside sophisticated data-structures (subgroup graphs, shift operations) and compressed representations (SLPs) to avoid exponential blowups. The results have implications for the complexity of fundamental decision problems in HNN-extensions and demonstrate a viable path to polynomial-time solutions via structural compression and normalizer-based reductions.

Abstract

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

Paper Structure

This paper contains 33 sections, 31 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. HNN extensions
  3. Our results
  4. Outline
  5. Model of computation and internal data representation
  6. Preliminaries: subgroup graphs
  7. Free groups and free monoids
  8. Data representation for words
  9. Subgroup graph
  10. A basis for $L(\Gamma)$
  11. Schreier graph
  12. Subgroup graph homomorphism
  13. Regularity, self-similarity, and shift operation
  14. Preliminaries: straight-line programs
  15. Definition of a straight-line program
  16. ...and 18 more sections

Key Result

Lemma 1.1

$w=1$ in $G\ast_\varphi t$ and $k\ge 1$$\ \ \Rightarrow\ \ w$ is not $t$-reduced.

Figures (1)

  • Figure 1: Concatenation of the paths $p(P_1,v_1)$ and $p(P_2,v_2)$.

Theorems & Definitions (54)

  • Lemma 1.1: Britton's lemma, brittonWordProblem1963
  • Corollary 1.2
  • Proposition 2.1: Kapovich_Miasnikov:2002
  • Theorem 2.2: Kapovich_Miasnikov:2002
  • proof
  • Proposition 2.3: Kapovich_Miasnikov:2002
  • Proposition 2.4: Kapovich_Miasnikov:2002
  • Theorem 2.5: Kapovich_Miasnikov:2002
  • Lemma 3.1
  • proof
  • ...and 44 more