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Membership problems for positive one-relator groups and one-relation monoids

Islam Foniqi, Robert D. Gray, Carl-Fredrik Nyberg-Brodda

TL;DR

The paper develops new undecidability phenomena for membership and word problems across one-relator monoids, positive one-relator groups, and inverse monoids, introducing trace-monoid embedding techniques and three key reduction avenues (inverse monoids, submonoid membership in positive one-relator groups, and principal right-ideal/rational subset membership). It constructs infinite families of undecidable cases, notably positive one-relator groups with undecidable submonoid membership and non-subspecial monoids with undecidable rational subset membership, and shows how these results imply undecidability for a broad class of one-relator monoids and for groups embedding trace monoids like $T(P_4)$. A central methodological advance is a general embedding framework (linking $T(P_4)$ into $\mathcal{L}$-classes) that transfers rational-subset undecidability from monoids to groups, complemented by explicit constructions such as $G_{m,n}$ and $R_{m,n}$ and by reductions to prefix membership in quasi-positive and two-relator inverse monoids. The work also connects these undecidability results to questions about surface groups and the word problem for inverse monoids, underscoring deep interplay between monoid/rational subset decidability and embeddings of RAAGs and trace monoids, and outlining several open directions (e.g., surface-group decidability and positive inverse monoid word problems).

Abstract

Motivated by approaches to the word problem for one-relation monoids arising from work of Adian and Oganesian (1987), Guba (1997), and Ivanov, Margolis and Meakin (2001), we study the submonoid and rational subset membership problems in one-relation monoids and in positive one-relator groups. We give the first known examples of positive one-relator groups with undecidable submonoid membership problem, and apply this to give the first known examples of one-relation monoids with undecidable submonoid membership problem. We construct several infinite families of one-relation monoids with undecidable submonoid membership problem, including examples that are defined by relations of the form $w=1$ but which are not groups, and examples defined by relations of the form $u=v$ where both of $u$ and $v$ are non-empty. As a consequence we obtain a classification of the right-angled Artin groups that can arise as subgroups of one-relation monoids. We also give examples of monoids with a single defining relation of the form $aUb = a$, and examples of the form $aUb=aVa$, with undecidable rational subset membership problem. We give a one-relator group defined by a freely reduced word of the form $uv^{-1}$ with $u, v$ positive words, in which the prefix membership problem is undecidable. Finally, we prove the existence of a special two-relator inverse monoid with undecidable word problem, and in which both the relators are positive words. As a corollary, we also find a positive two-relator group with undecidable prefix membership problem. In proving these results, we introduce new methods for proving undecidability of the rational subset membership problem in monoids and groups, including by finding suitable embeddings of certain trace monoids.

Membership problems for positive one-relator groups and one-relation monoids

TL;DR

The paper develops new undecidability phenomena for membership and word problems across one-relator monoids, positive one-relator groups, and inverse monoids, introducing trace-monoid embedding techniques and three key reduction avenues (inverse monoids, submonoid membership in positive one-relator groups, and principal right-ideal/rational subset membership). It constructs infinite families of undecidable cases, notably positive one-relator groups with undecidable submonoid membership and non-subspecial monoids with undecidable rational subset membership, and shows how these results imply undecidability for a broad class of one-relator monoids and for groups embedding trace monoids like . A central methodological advance is a general embedding framework (linking into -classes) that transfers rational-subset undecidability from monoids to groups, complemented by explicit constructions such as and and by reductions to prefix membership in quasi-positive and two-relator inverse monoids. The work also connects these undecidability results to questions about surface groups and the word problem for inverse monoids, underscoring deep interplay between monoid/rational subset decidability and embeddings of RAAGs and trace monoids, and outlining several open directions (e.g., surface-group decidability and positive inverse monoid word problems).

Abstract

Motivated by approaches to the word problem for one-relation monoids arising from work of Adian and Oganesian (1987), Guba (1997), and Ivanov, Margolis and Meakin (2001), we study the submonoid and rational subset membership problems in one-relation monoids and in positive one-relator groups. We give the first known examples of positive one-relator groups with undecidable submonoid membership problem, and apply this to give the first known examples of one-relation monoids with undecidable submonoid membership problem. We construct several infinite families of one-relation monoids with undecidable submonoid membership problem, including examples that are defined by relations of the form but which are not groups, and examples defined by relations of the form where both of and are non-empty. As a consequence we obtain a classification of the right-angled Artin groups that can arise as subgroups of one-relation monoids. We also give examples of monoids with a single defining relation of the form , and examples of the form , with undecidable rational subset membership problem. We give a one-relator group defined by a freely reduced word of the form with positive words, in which the prefix membership problem is undecidable. Finally, we prove the existence of a special two-relator inverse monoid with undecidable word problem, and in which both the relators are positive words. As a corollary, we also find a positive two-relator group with undecidable prefix membership problem. In proving these results, we introduce new methods for proving undecidability of the rational subset membership problem in monoids and groups, including by finding suitable embeddings of certain trace monoids.
Paper Structure (11 sections, 31 theorems, 96 equations, 1 figure)

This paper contains 11 sections, 31 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction and summary of results
  2. Preliminaries
  3. Membership Problems in Positive One-relator Groups
  4. Positive one-relator groups and surface groups
  5. Prefix Membership Problems
  6. Membership Problems in One-relation Monoids
  7. Special and subspecial monoids
  8. Non-subspecial monoids
  9. Undecidability in monadic one-relation monoids
  10. An application to the rational subset membership problem in groups
  11. The word problem for positive special inverse monoids

Key Result

Lemma 3.2

The group $G_{m,n}$ has decidable submonoid membership problem if and only if $m =1$ or $n=1$. Furthermore, if $m, n \geq 2$ then $G_{m,n}$ contains a fixed finitely generated submonoid in which membership is undecidable.

Figures (1)

  • Figure 1: A summary of the main results of this article and how they relate to the three approaches to the word problem for one-relation monoids given by reduction results of (i) Ivanov, Margolis and Meakin Ivanov2001, (ii) Guba Guba1997, and (iii) Adian and Oganesian Adian1966Adian1976Adian1987. The arrows indicate implication of decidability. The problems in red are all proved to be undecidable in this article in the results listed in the corresponding boxes. The problems in white boxes are all open.

Theorems & Definitions (61)

  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • ...and 51 more