On character of points in the Higson corona of a metric space

TL;DR

This paper characterizes the minimal character of points in the Higson corona of an unbounded metric space, linking it to set-theoretic invariants and coarse geometric properties.

Contribution

It establishes a precise relationship between the minimal character of Higson corona points and set-theoretic invariants, and characterizes when a space is coarsely equivalent to the Cantor macro-cube.

Findings

Minimal character equals  u if asymptotically isolated balls exist.

Minimal character equals  max{ u, d} otherwise.

Under  u< d, spaces with bounded geometry are characterized by corona dimension and minimal character.

Abstract

We prove that for an unbounded metric space $X$, the minimal character $m\chi(\check X)$ of a point of the Higson corona $\check X$ of $X$ is equal to $\mathfrak u$ if $X$ has asymptotically isolated balls and to $\max\{\mathfrak u,\mathfrak d\}$ otherwise. This implies that under $\mathfrak u<\mathfrak d$ a metric space $X$ of bounded geometry is coarsely equivalent to the Cantor macro-cube $2^{<\IN}$ if and only if $\dim(\check X)=0$ and $m\chi(\check X)=\mathfrak d$. This contrasts with a result of Protasov saying that under CH the coronas of any two asymptotically zero-dimensional unbounded metric separable spaces are homeomorphic.