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 $L$ 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.