On epiC groups over language class C

TL;DR

This paper introduces epiC groups, a new class linking group theory and formal language classes, and explores their properties, closure under various operations, and relation to the solvable word problem.

Contribution

It defines epiC groups within the Chomsky hierarchy, proves their invariance under generating set choice, and establishes their closure properties and connection to the word problem.

Findings

epiC groups are invariant under generating set choice

epiC groups are closed under finite index overgroups, extensions, and graph products

epiRegular groups are closed under finite index subgroups

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.