# Applications of L systems to group theory

## TL;DR

This paper explores how L systems, a type of formal grammar, can generate various complex languages in group theory, revealing new connections and classifications within formal language hierarchies.

## Contribution

It demonstrates that many group-theoretic languages are in fact EDT0L or ET0L, expanding the understanding of their formal language classifications.

## Key findings

- Languages of primitives in free groups on two generators are EDT0L.
- Languages of primitives in higher rank free groups are only context-sensitive.
- Existence of EDT0L and ET0L languages with intermediate growth rates.

## Abstract

L systems generalise context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives in the free group on two generators, the Bridson-Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk's group can be generated by L systems. To complement the result on primitives in free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L and ET0L languages of intermediate growth.