The submonoid and rational subset membership problems for graph groups
摘要
We show that the membership problem in a finitely generated submonoid of a graph group (also called a right-angled Artin group or a free partially commutative group) is decidable if and only if the independence graph (commutation graph) is a transitive forest. As a consequence we obtain the first example of a finitely presented group with a decidable generalized word problem that does not have a decidable membership problem for finitely generated submonoids. We also show that the rational subset membership problem is decidable for a graph group if and only if the independence graph is a transitive forest, answering a question of Kambites, Silva, and the second author. Finally we prove that for certain amalgamated free products and HNN-extensions the rational subset and submonoid membership problems are recursively equivalent. In particular, this applies to finitely generated groups with two or more ends that are either torsion-free or residually finite.
引用
@article{arxiv.math/0608768,
title = {The submonoid and rational subset membership problems for graph groups},
author = {Markus Lohrey and Benjamin Steinberg},
journal= {arXiv preprint arXiv:math/0608768},
year = {2007}
}