On metric spaces with given transfinite asymptotic dimensions

TL;DR

This paper constructs metric spaces with specified transfinite asymptotic dimensions, advancing understanding of the structure and properties of such spaces in geometric group theory.

Contribution

It introduces a method to explicitly construct metric spaces with prescribed transfinite asymptotic and complementary-finite asymptotic dimensions.

Findings

Constructed spaces for every countable ordinal with matching dimensions

Demonstrated the existence of spaces with complex transfinite asymptotic structures

Provided new tools for analyzing large-scale geometric properties

Abstract

For every countable ordinal number $\xi$, we construct a metric space $X_{\xi}$ whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both $\xi$.