Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Groups with a finite Busemann boundary are virtually cyclic

Corentin Bodart, Liran Ron-George, Ariel Yadin

TL;DR

The paper analyzes how the size of the metric-functional (Busemann) boundary of a finitely generated group, viewed through left-invariant metrics and Cayley graphs, characterizes the group's algebraic structure. It introduces annihilators to relate boundary functionals to group elements and proves that if any Cayley graph has a finite set of Busemann points, the group must be virtually cyclic (virtually-$\mathbb{Z}$). The work reduces the problem to the virtually abelian case, proving that in rank $\ge 2$ virtually abelian groups the Busemann boundary is infinite, while establishing finiteness consequences and local finiteness properties of the annihilator. Collectively with prior results, it completes the characterization of groups with finite metric-functional boundaries, tying geometric compactifications to finite-index cyclic subgroups. The methods combine geodesic limits, convex geometry via extreme points, and a careful analysis of the annihilator's structure to bridge geometry and group theory.

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.

Groups with a finite Busemann boundary are virtually cyclic

TL;DR

The paper analyzes how the size of the metric-functional (Busemann) boundary of a finitely generated group, viewed through left-invariant metrics and Cayley graphs, characterizes the group's algebraic structure. It introduces annihilators to relate boundary functionals to group elements and proves that if any Cayley graph has a finite set of Busemann points, the group must be virtually cyclic (virtually-). The work reduces the problem to the virtually abelian case, proving that in rank virtually abelian groups the Busemann boundary is infinite, while establishing finiteness consequences and local finiteness properties of the annihilator. Collectively with prior results, it completes the characterization of groups with finite metric-functional boundaries, tying geometric compactifications to finite-index cyclic subgroups. The methods combine geodesic limits, convex geometry via extreme points, and a careful analysis of the annihilator's structure to bridge geometry and group theory.

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.

Paper Structure

This paper contains 8 sections, 16 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Main results
  4. Annihilators
  5. Geodesics
  6. The annihilator is locally finite
  7. Virtually abelian groups
  8. Proof of main results

Key Result

Theorem 1.1

Let $G$ be a finitely generated, infinite group. The following are equivalent.

Figures (3)

  • Figure 1: The Cayley graph of $\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}$, with $N$ in red.
  • Figure 2: Finite geodesics $\alpha^{(n)}$ with endpoints in $N$ limiting to an infinite geodesic $\beta$ satisfying Equation (\ref{['eq:unit_speed']}).
  • Figure 3: The geodesics $\alpha^{(n)}$, with their "bend" translated back to $1$ so that Equation (\ref{['eq:unit_speed']}) fails from the start.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • proof
  • Theorem 1.4
  • Lemma 1.5
  • Proposition 1.6
  • Lemma 1.7
  • proof
  • Corollary 1.8
  • ...and 25 more