Uniquely labelled geodesics of Coxeter groups

TL;DR

This paper introduces the concept of uniquely labelled geodesics in Coxeter groups, providing enumeration formulas for finite types and exploring properties of infinite Coxeter groups, including examples and geometric descriptions.

Contribution

It defines uniquely labelled geodesics in Coxeter groups, derives generating functions for finite types, and analyzes their properties in infinite groups, including existence and geometric structure.

Findings

Generated explicit formulas for u.l.g.'s in type A_n

Proved finite length of u.l.g.'s with finite branching in infinite groups

Identified infinite u.l.g.'s in specific infinite Coxeter groups like _6

Abstract

Studying geodesics in Cayley graphs of groups has been a very active area of research over the last decades. We introduce the notion of a uniquely labelled geodesic, abbreviated with u.l.g. These will be studied first in finite Coxeter groups of type $A_n$. Here we introduce a generating function, and hence are able to precisely describe how many u.l.g.'s we have of a certain length and with which label combination. These results generalize several results about unique geodesics in Coxeter groups. In the second part of the paper, we expand our investigation to infinite Coxeter groups described by simply laced trees. We show that any u.l.g. of finite branching index has finite length. We use the example of the group $\widetilde{D}_6$ to show the existence of infinite u.l.g.'s in groups which do not have any infinite unique geodesics. We conclude by exhibiting a detailed description of the geometry of such u.l.g.'s and their relation to each other in the group $\widetilde{D}_6$.