Geodesic words in central extension of the Klein bottle group

TL;DR

This paper characterizes geodesic words in a central extension of the Klein bottle group, proving the absence of dead ends and providing conditions for their non-existence, advancing understanding of the group's geometric structure.

Contribution

It provides a complete description of geodesic words and introduces a new condition ensuring no dead ends in the group.

Findings

Complete description of geodesic words

Proof of no dead ends in the group

New sufficient condition for absence of dead ends

Abstract

In this work, we give a complete description of the language of geodesic words for a central extension of the fundamental group of the Klein Bottle with respect to the standard two-element generating set. Besides, we prove that there are no dead ends in the group. Finally, we give a new sufficient condition for the absence of dead ends.