Full box spaces of free groups

TL;DR

This paper studies the coarse geometric properties of full box spaces of free groups and abelian groups, showing non-equivalence under certain conditions related to the number of generators and group rank.

Contribution

It establishes new distinctions between full box spaces of free groups with different generators and between abelian groups and 2-generated groups in coarse geometry.

Findings

Full box spaces of free groups with different numbers of generators are not coarsely equivalent when the number of generators differs significantly.

The full box space of al^n is not coarsely equivalent to that of any 2-generated group for n
0.

Full box spaces of free groups and al^n exhibit distinct coarse geometric properties.

Abstract

In this paper we investigate full box spaces and coarse equivalences between them. We do this in two parts. In part one we compare the full box spaces of free groups on different numbers of generators. In particular the full box space of a free group $F_k$ is not coarsely equivalent to the full box space of a free group $F_d$, if $d\ge 8k+10$. In part two we compare $\Box_f\mathbb{Z}^n$ to the full box spaces of $2$-generated groups. In particular we prove that the full box space of $\mathbb{Z}^n$ is not coarsely equivalent to the full box space of any $2$-generated group, if $n\ge 3$.