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On linguistic subsets of groups and monoids

André Carvalho, Carl-Fredrik Nyberg-Brodda

TL;DR

This work develops a general framework to study language-theoretic subsets of groups and monoids via classes of languages $\mathbf{C}$, introducing $\mathbf{C}^\forall$- and $\mathbf{C}^\exists$-flatness and their relative variants. It establishes fundamental closure properties, a structure theorem for $\mathbf{C}^\bullet$-subsets, and a suite of characterizations for specific language classes, notably proving $\mathbf{1C}^\forall$-flatness iff virtual cyclicity and, for $\mathbf{RE}$, that $\mathbf{RE}^\forall$-flatness coincides with recursive presentation (Higman). The paper also provides counterexamples showing that flatness is not in general equivalent to having the word problem in $\mathbf{C}$, with explicit results for $\mathbf{REC}$, $\mathbf{CS}$, and $\mathbf{pCF}$, and constructs using Tarski monsters to separate $\mathbf{RE}^\exists$-flatness from $\mathbf{RE}^\forall$-flatness. Together, these results illuminate a nuanced landscape linking language theory, subgroup membership problems, and algorithmic decidability in group theory, including novel closure properties for epi-$\mathbf{C}$-groups and finite-index subgroups.

Abstract

We study subsets of groups and monoids defined by language-theoretic means, generalizing the classical approach to the word problem. We expand on results by Herbst from 1991 to a more general setting, and for a class of languages $\mathbf{C}$ we define the classes of $\mathbf{C}^\forall$-flat and $\mathbf{C}^\exists$-flat groups. We prove several closure results for these classes of groups, prove a connection with the word problem, and characterize $\mathbf{C}^\forall$-flat groups for several classes of languages. In general, we prove that the class of $\mathbf{C}^\forall$-flat groups is a strict subclass of the class of groups with word problem in $\mathbf{C}$, including for the class $\mathbf{REC}$ of recursive languages, for which $\mathbf{C}^\forall$-flatness for a group resp. monoid is proved to be equivalent to the decidability of the subgroup membership problem resp. the submonoid membership problem. We provide a number of examples, including the Tarski monsters of Ol'shanskii, showing the difficulty of characterizing $\mathbf{C}^\exists$-flat groups. As an application of our general methods, we also prove in passing that if $\mathbf{C}$ is a full semi-$\mathrm{AFL}$, then the class of epi-$\mathbf{C}$ groups is closed under taking finite index subgroups. This answers a question recently posed by Al Kohli, Bleak & Elliott.

On linguistic subsets of groups and monoids

TL;DR

This work develops a general framework to study language-theoretic subsets of groups and monoids via classes of languages , introducing - and -flatness and their relative variants. It establishes fundamental closure properties, a structure theorem for -subsets, and a suite of characterizations for specific language classes, notably proving -flatness iff virtual cyclicity and, for , that -flatness coincides with recursive presentation (Higman). The paper also provides counterexamples showing that flatness is not in general equivalent to having the word problem in , with explicit results for , , and , and constructs using Tarski monsters to separate -flatness from -flatness. Together, these results illuminate a nuanced landscape linking language theory, subgroup membership problems, and algorithmic decidability in group theory, including novel closure properties for epi--groups and finite-index subgroups.

Abstract

We study subsets of groups and monoids defined by language-theoretic means, generalizing the classical approach to the word problem. We expand on results by Herbst from 1991 to a more general setting, and for a class of languages we define the classes of -flat and -flat groups. We prove several closure results for these classes of groups, prove a connection with the word problem, and characterize -flat groups for several classes of languages. In general, we prove that the class of -flat groups is a strict subclass of the class of groups with word problem in , including for the class of recursive languages, for which -flatness for a group resp. monoid is proved to be equivalent to the decidability of the subgroup membership problem resp. the submonoid membership problem. We provide a number of examples, including the Tarski monsters of Ol'shanskii, showing the difficulty of characterizing -flat groups. As an application of our general methods, we also prove in passing that if is a full semi-, then the class of epi- groups is closed under taking finite index subgroups. This answers a question recently posed by Al Kohli, Bleak & Elliott.

Paper Structure

This paper contains 15 sections, 35 theorems, 26 equations.

Table of Contents

  1. Preliminaries
  2. Language-theoretic background
  3. Groups and languages
  4. General properties of $\mathbf{C}$-subsets
  5. $\mathbf{C}^\bullet$-subsets
  6. Relative $\mathbf{C}^\bullet$-subsets
  7. $\mathbf{C}^\bullet$-flatness
  8. Closure properties of $\mathbf{C}^\bullet$-flat monoids and groups
  9. Characterizing $\mathbf{C}^\bullet$-flat groups for specific classes
  10. One-counter flatness
  11. Recursive and context-sensitive flatness
  12. Recursively enumerable flatness
  13. Open problems and questions
  14. Connections with conjugacy classes
  15. List of open problems

Key Result

Theorem 1.1

Let $G = \langle A \mid R \rangle$ be a finitely generated group. Then we have:

Theorems & Definitions (63)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2: Anisimov & Seifert Anisimov1975
  • Proposition 1.3
  • Lemma 2.1: Herbst Herbst1991
  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Herbst Herbst1991
  • Lemma 2.4: Herbst Herbst1991
  • ...and 53 more