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Abstract
We prove the following results: (1) There is a onerelator inverse monoid Inv⟨Aw=1⟩ with undecidable word problem; and (2) There are onerelator groups with undecidable submonoid membership problem. The second of these results is proved by showing that for any finite forest the associated rightangled Artin group embeds into a onerelator 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 onerelator 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 onerelator group and submonoid in which membership is undecidable from (2) to construct a onerelator inverse monoid Inv⟨Aw=1⟩ with undecidable word problem. Furthermore, this method allows the construction of an Eunitary onerelator 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).
Original language  English 

Pages (fromto)  9871008 
Number of pages  22 
Journal  Inventiones Mathematicae 
Volume  219 
Issue number  3 
Early online date  9 Sep 2019 
DOIs  
Publication status  Published  Mar 2020 
Keywords
 20F05
 20F10
 20F36
 20M05
 20M18
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Robert Gray
 School of Mathematics  Reader in Pure Mathematics
 Algebra and Combinatorics  Member
 Logic  Member
Person: Research Group Member, Academic, Teaching & Research
Projects
 1 Finished