A nonstandard invariant of coarse spaces

Takuma Imamura – GJM, Volume 5, Issue 1 (2020), 1-8.

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octobre 28, 2020
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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. [15] 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 $\left(X,\xi\right)$ .

Categories: Issue1, 2020

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Received: September 26, 2019.
Accepted: February 22, 2020.
Published online: April 3, 2020.

Authors:

Takuma Imamura
RIMS, Kyoto University, Kitashirakawa Oiwake-cho,
Sakyo-ku, Kyoto 606-8502, Japan.

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