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

2025-10-04

AI Summary

This paper investigates the word problem complexity for HNN extensions of free groups under the condition that the associated subgroups are equal and of finite index. While classical theory restricts attention to normal subgroups, we extend polynomial-time solvability to non-normal subgroups of finite index satisfying a specific isomorphism condition. Our approach integrates combinatorial group theory with computational complexity analysis, leveraging the geometric structure of free groups and the boundedness of subgroup index to construct a polynomial-time algorithm for solving the word problem. The main contribution is the first proof that the word problem for such HNN extensions admits a polynomial-time upper bound. This significantly broadens the scope of algorithmic HNN extension theory and establishes a novel paradigm for algorithmic studies of non-normal subgroups in group theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Let $G=Fast_varphi t$ 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 $varphi:H_1 o H_2$ satisfies an additional condition described in Section 5.

Problem

• Solving polynomial time word problem for HNN extensions
• Extending results to non-normal equal subgroups
• Analyzing free group extensions with finite index

Innovation

• HNN extensions with equal associated subgroups
• Polynomial time word problem solution
• Extended to non-normal subgroups with conditions