There is no universal proper metric spaces for asymptotic dimension 1

TL;DR

This paper proves that for any dimension n ≥ 1, no single proper metric space can serve as a universal space for all spaces with asymptotic dimension n, resolving a question in geometric group theory.

Contribution

It establishes the non-existence of a universal proper metric space for asymptotic dimension n ≥ 1, answering an open question in the field.

Findings

No universal proper metric space exists for asymptotic dimension n ≥ 1.

The result applies to all n ≥ 1, including the case n=1.

This advances understanding of the structure of metric spaces in geometric group theory.

Abstract

Answering a question of Ma, Siegert, and Dydak we show that there is no universal proper metric space for the asymptotic dimension $n\ge1$.