Groups with a finite Busemann boundary are virtually cyclic

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 .