Complexity of word problems for HNN-extensions

TL;DR

This paper investigates the computational complexity of the word problem in HNN-extensions, showing polynomial-time solutions for certain hyperbolic group extensions and establishing reductions to compressed word problems.

Contribution

It demonstrates that the word problem for ascending HNN-extensions of hyperbolic groups with cyclic subgroups can be solved in polynomial time, extending to graphs of groups.

Findings

Word problem for ascending HNN-extensions reduces to compressed word problem.

Polynomial-time solution for HNN-extensions of hyperbolic groups with cyclic subgroups.

Extension to fundamental groups of graphs of groups with hyperbolic vertex groups.

Abstract

The computational complexity of the word problem in HNN-extension of groups is studied. HNN-extension is a fundamental construction in combinatorial group theory. It is shown that the word problem for an ascending HNN-extension of a group H is logspace reducible to the so-called compressed word problem for H. The main result of the paper states that the word problem for an HNN-extension of a hyperbolic group H with cyclic associated subgroups can be solved in polynomial time. This result can be easily extended to fundamental groups of graphs of groups with hyperbolic vertex groups and cyclic edge groups.