Horofunctions on graphs of linear growth

TL;DR

This paper proves that graphs with linear growth have finitely many horofunctions, leading to a simple proof that such groups are virtually cyclic, connecting geometric properties with algebraic group structure.

Contribution

It establishes a finite horofunctions property for linear growth graphs and simplifies the proof that these groups are virtually cyclic.

Findings

Linear growth graphs have finitely many horofunctions.

Finitely generated infinite groups of linear growth are virtually cyclic.

Provides a concise proof linking geometric and algebraic properties.

Abstract

We prove that a linear growth graph has finitely many horofunctions. This provides a short and simple proof that any finitely generated infinite group of linear growth is virtually cyclic.