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On epiC groups over language class C

Raad Al Kohli, Collin Bleak, Luna Elliott

Abstract

We introduce a new framework linking group theory and formal language theory which generalizes a number of ways these topics have been linked in the past. For a language class C in the Chomsky hierarchy, we say a group is epiC if it admits a language $L$ over a finite (monoidal) generating set $X \subseteq G$ in the class C such that the image of L under the evaluation map is $G \setminus \{1_G\}$. We provide some examples of epiC groups and prove that the property of being epiC is not dependent on the generating set chosen. We also prove that epiC groups are closed under passage to finite index overgroups, taking extensions, and taking graph products of finitely many groups. Furthermore, we prove that epiRegular groups are closed under passage to finite index subgroups. Finally, we provide a characterization of the property of having solvable word problem within the framework of epiC groups.

On epiC groups over language class C

Abstract

We introduce a new framework linking group theory and formal language theory which generalizes a number of ways these topics have been linked in the past. For a language class C in the Chomsky hierarchy, we say a group is epiC if it admits a language over a finite (monoidal) generating set in the class C such that the image of L under the evaluation map is . We provide some examples of epiC groups and prove that the property of being epiC is not dependent on the generating set chosen. We also prove that epiC groups are closed under passage to finite index overgroups, taking extensions, and taking graph products of finitely many groups. Furthermore, we prove that epiRegular groups are closed under passage to finite index subgroups. Finally, we provide a characterization of the property of having solvable word problem within the framework of epiC groups.
Paper Structure (15 sections, 27 theorems, 30 equations)

This paper contains 15 sections, 27 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Languages and groups: some formal definitions
  4. Independence of generating sets.
  5. Closure Properties
  6. Finite index overgroups
  7. Extensions
  8. Finite index subgroups
  9. Graph products of groups
  10. Autostackable groups
  11. Non-examples
  12. Word problems
  13. Continuum many non-epiRE 2-generated groups
  14. Continuum many epiregular groups
  15. Questions

Key Result

Theorem A

Suppose C $\in\{Regular, CF, CS, RE\}$ and $G$ is a group with finite subsets $X$ and $Y$ where both $X$ and $Y$ generate $G$ as a monoid. If $G$ is epiC with respect to $X$ then $G$ is epiC with respect to $Y$.

Theorems & Definitions (57)

  • Theorem A
  • Theorem B
  • Corollary 2.1
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem G
  • Definition 2.1: Word and coword problem
  • Remark 2.2
  • ...and 47 more