# Asymptotic property C of the countable direct sum of uniformly discrete $0$-hyperbolic spaces

## TL;DR

The paper proves that the countable direct sum of 0-hyperbolic, D-discrete metric spaces has asymptotic property C, extending previous results to a broader class including free groups of finite rank.

## Contribution

It introduces a new definition of the direct sum of pointed metric spaces and proves that such sums of 0-hyperbolic, D-discrete spaces possess asymptotic property C.

## Key findings

- Countable direct sums of 0-hyperbolic, D-discrete spaces have asymptotic property C.
- Includes the case of countable sums of free groups of finite rank.
- Generalizes Yamauchi's result on sums of integers.

## Abstract

We define the direct sum of a countable family of pointed metric spaces in a way resembling the direct sum of groups. Then we prove that if a family consists of $0$-hyperbolic (in the sense of Gromov) and $D$-discrete spaces, then its direct sum has asymptotic property C. The main example is a countable direct sum of free groups of (possibly varying) finite rank. This is a generalization of T. Yamauchi's result concernig the countable direct sum of the integers.