Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups

Gray, Robert (2020) Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups. Inventiones Mathematicae, 219 (3). pp. 987-1008. ISSN 0020-9910

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Abstract

We prove the following results: (1) There is a one-relator inverse monoid Inv⟨A|w=1⟩ with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The second of these results is proved by showing that for any finite forest the associated right-angled Artin group embeds into a one-relator group. Combining this with a result of Lohrey and Steinberg (J Algebra 320(2):728–755, 2008), we use this to prove that there is a one-relator group containing a fixed finitely generated submonoid in which the membership problem is undecidable. To prove (1) a new construction is introduced which uses the one-relator group and submonoid in which membership is undecidable from (2) to construct a one-relator inverse monoid Inv⟨A|w=1⟩ with undecidable word problem. Furthermore, this method allows the construction of an E-unitary one-relator inverse monoid of this form with undecidable word problem. The results in this paper answer a problem originally posed by Margolis et al. (in: Semigroups and their applications, Reidel, Dordrecht, pp. 99–110, 1987).

Item Type: Article
Uncontrolled Keywords: 20f05,20f10,20f36,20m05,20m18,free-products,identity problem
Faculty \ School: Faculty of Science > School of Mathematics
Related URLs:
Depositing User: LivePure Connector
Date Deposited: 09 Sep 2019 15:30
Last Modified: 13 Jul 2020 23:54
URI: https://ueaeprints.uea.ac.uk/id/eprint/72132
DOI: 10.1007/s00222-019-00920-2

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