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

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 from 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, Meakin and Stephen in 1987.

A tribute to John Meakin on the occasion of his 75th birthday

“…In [11] they develop methods to solve the word problem for this and related monoids. Gray also proved that the word problem for one-relator monoids with a non-reduced relator w = 1 can have an undecidable word problem [14], a spectacular result, considering the Magnus Theorem on one-relator groups and Adian's decidability result for one relator monoids with relator w = 1 . Thus the problem for one-relator inverse monoids of the form w = 1 remains active and with deep results.…”

New results on the prefix membership problem for one-relator groups

“…This strongly motivates the study of special inverse monoids and their word problems, which is also interesting in its own right, given the prevalence of inverse semigroups and their combinatorial and geometric aspects in various areas of mathematics (see [24]). However, a recent surprising result of Gray [10] shows that the word problem for one-relator special inverse monoids in complete generality is undecidable.…”

“…For example, it is still unknown whether the subgroup membership problem-also called the generalised word problem-is decidable for one-relator groups. However, there exist one-relator groups in which the submonoid membership problem (and thus the more general rational subset membership problem [25]) is undecidable [10]. The one-relator group with undecidable submonoid membership problem given in [10] is an HNN extension of Z × Z with respect to an isomorphism mapping one of the natural copies of Z to the other.…”

“…However, there exist one-relator groups in which the submonoid membership problem (and thus the more general rational subset membership problem [25]) is undecidable [10]. The one-relator group with undecidable submonoid membership problem given in [10] is an HNN extension of Z × Z with respect to an isomorphism mapping one of the natural copies of Z to the other. So in general the submonoid and rational subset membership problems are not well-behaved under the HNN extension construction, and similarly for free products with amalgamation.…”

“…In the last section of the paper we present a result of a different flavour which says something about the limits of what we should hope to be able to prove about the prefix membership problem in one-relator groups. Specifically, by modifying the construction from [10], we will show in Theorem 8.2 that there is a finite alphabet X and a reduced word w ∈ (X ∪ X −1 ) * such that Gp X | w = 1 has undecidable prefix membership problem. Hence if [5,Question 13.10] has a positive answer then the cyclically reduced hypothesis will need to be used.…”