# Counting coarse subsets of a countable group

## TL;DR

This paper establishes that for any countable group, there are uncountably many classes of subsets that are coarsely equivalent, highlighting the rich structure of coarse geometry within such groups.

## Contribution

It proves that the number of coarse equivalence classes of subsets in any countable group is maximal, specifically $2^{ ext{omega}}$, which was previously unknown.

## Key findings

- Uncountably many coarse equivalence classes exist in countable groups.
- The number of classes is $2^{ ext{omega}}$, the cardinality of the continuum.
- This result reveals the complexity of coarse geometric structures in countable groups.

## Abstract

For every countable group $G$, there are $2^{\omega}$ distinct classes of coarsely equivalent subsets of $G$.