Groups with a finite Busemann boundary are virtually cyclic

2025/11/25 by Corentin Bodart, Liran Ron-George, Liran Ron‐George +4
Mathematics · #math.GR #math.MG

paper · pdf · doi:10.48550/arxiv.2511.20495

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.

Citations

• Hahn-Banach for metric functionals and horofunctions
• On the horofunction boundary of discrete Heisenberg group
• Busemann points of infinite graphs

Related

• Groups with finitely many Busemann points
• Busemann Points of Infinite Graphs
• Busemann points of infinite graphs
• Being Cayley automatic is closed under taking wreath product with\n virtually cyclic groups
• The Geometry of Cycles in the Cayley Diagram of a Group