Busemann points of Artin groups of dihedral type

TL;DR

This paper investigates the horofunction boundary of dihedral Artin groups with different generating sets, identifying Busemann points and characterizing geodesics to understand their geometric structure.

Contribution

It provides a detailed description of the horoboundary and Busemann points for dihedral Artin groups with both usual and dual generators, and characterizes geodesics for the dual case.

Findings

All boundary points are Busemann points with dual generators.

Not all boundary points are Busemann points with usual generators.

Geodesic growth series are computed for dual generators.

Abstract

We study the horofunction boundary of an Artin group of dihedral type with its word metric coming from either the usual Artin generators or the dual generators. In both cases, we determine the horoboundary and say which points are Busemann points, that is the limits of geodesic rays. In the case of the dual generators, it turns out that all boundary points are Busemann points, but this is not true for the Artin generators. We also characterise the geodesics with respect to the dual generators, which allows us to calculate the associated geodesic growth series.