Groups with a finite Busemann boundary are virtually cyclic

TL;DR

This paper proves that groups with a Cayley graph having finitely many Busemann points are virtually cyclic, completing the classification of groups with finite metric-functional boundaries.

Contribution

It establishes a new characterization linking finite Busemann boundaries to virtually cyclic groups, introducing the concept of annihilators.

Findings

Groups with finitely many Busemann points are virtually cyclic

Complete classification of groups with finite metric-functional boundaries

Introduces the notion of annihilators

Abstract

This note is a continuation of the study of the relationship between the geometry of Cayley graphs and the size of its metric-functional boundary. We show that, if there exists a Cayley graph with finitely many Busemann points, then the underlying group is virtually cyclic. Together with previous works, this completes the full characterization of groups with finite metric-functional boundaries. The main new notion introduced is that of annihilators.