Groups with a finite Busemann boundary are virtually cyclic
arXiv:2511.20495 · doi:10.1112/blms.70462
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.
v2: Added Proposition 3.3, fixed minors mistakes. Accepted to Bulletin of the LMS. 18 pages, 3 figures