A nonstandard invariant of coarse spaces

TL;DR

This paper introduces a new set-valued invariant for pointed coarse spaces using nonstandard analysis, compares it with a standard invariant, and proves properties like invariance under coarse equivalence and surjectivity of a related transformation.

Contribution

It constructs a nonstandard set-valued invariant for coarse spaces, establishes its invariance, and compares it with a standard invariant via a natural transformation.

Findings

The invariant is invariant under coarse equivalence.

A sufficient condition for the invariant to have cardinality ≤1 is provided.

The natural transformation from the standard to the nonstandard invariant is surjective for proper geodesic spaces.

Abstract

We construct a set-valued invariant $\iota\left(X,\xi\right)$ of pointed coarse spaces $\left(X,\xi\right)$ by using nonstandard analysis. The invariance under coarse equivalence is established. A sufficient condition for the invariant to be of cardinality $\leq1$ is provided. Miller et al. and subsequent researchers have introduced a similar but standard set-valued coarse invariant $\sigma\left(X,\xi\right)$ of pointed metric spaces $\left(X,\xi\right)$. In order to compare these two invariants, we construct a natural transformation $\omega_{\left(X,\xi\right)}$ from $\sigma\left(X,\xi\right)$ to $\iota\left(X,\xi\right)$. The surjectivity of $\omega_{\left(X,\xi\right)}$ is proved for all proper geodesic spaces $\left(X,\xi\right)$.