Busemann Points of Infinite Graphs

Abstract

We provide a geometric condition which determines whether or not every point on the metric boundary of a graph with the standard path metric is a Busemann point, that is it is the limit point of a geodesic ray. We apply this and a related condition to investigate the structure of the metric boundary of Cayley graphs. We show that every metric boundary point of the Cayley graph of a finitely generated Abelian group is a Busemann point, but groups such as the braid group and the discrete Heisenberg group have boundary points of the Cayley graph which are not Busemann points when equipped with their usual generators.