# Coarse free products

## TL;DR

This paper introduces a new concept of free products for coarse spaces, extending group free products, and demonstrates that these preserve key coarse properties and provides bounds on their dimension.

## Contribution

It defines coarse free products, generalizes group free products to coarse spaces, and analyzes their properties and dimensions.

## Key findings

- Free products preserve coarse property C.
- Free products preserve finite coarse decomposition complexity.
- An upper bound on the dimension of coarse free products is established.

## Abstract

We define a notion of free product for coarse spaces that generalizes the corresponding notion of a free product for groups. We show that free products preserve coarse properties such as coarse property C, finite coarse decomposition complexity, and coarse property A. We also give an upper bound estimate on the dimension of a coarse free product in terms of the dimension of its factors.