# Piecewise excluding geodesic languages

## TL;DR

This paper investigates the properties of geodesic languages in groups, showing that certain non-abelian groups can have piecewise excluding geodesic languages, expanding understanding beyond previously known abelian cases.

## Contribution

It demonstrates that non-abelian groups can possess piecewise excluding geodesic languages and characterizes the quaternion group as uniquely having this property for all generating sets.

## Key findings

- Non-abelian groups can have piecewise excluding geodesic languages.
- The quaternion group is the only non-abelian 2-generator group with this property for all generating sets.
- Some virtually abelian groups lack piecewise excluding geodesic languages for any generating set.

## Abstract

The complexity of a geodesic language has connections to algebraic properties of the group. Gilman, Hermiller, Holt, and Rees show that a finitely generated group is virtually free if and only if its geodesic language is locally excluding for some finite inverse-closed generating set. The existence of such a correspondence and the result of Hermiller, Holt, and Rees that finitely generated abelian groups have piecewise excluding geodesic language for all finite inverse-closed generating sets motivated our work. We show that a finitely generated group with piecewise excluding geodesic language need not be abelian and give a class of infinite non-abelian groups which have piecewise excluding geodesic languages for certain generating sets. The quaternion group is shown to be the only non-abelian 2-generator group with piecewise excluding geodesic language for all finite inverse-closed generating sets. We also show that there are virtually abelian groups with geodesic languages which are not piecewise excluding for any finite inverse-closed generating set.