The submonoid and rational subset membership problems for graph groups
نویسندگان
چکیده
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 [24]. 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.
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