On the Structure of Asymptotic Space of the Lobachevsky Plane

TL;DR

This paper characterizes the asymptotic space of the Lobachevsky plane using Nonstandard Analysis, revealing it to be an R-tree with diverse nonisometric variants.

Contribution

It provides an exhaustive description of the asymptotic spaces of the Lobachevsky plane, highlighting their structure and variety.

Findings

Asymptotic space of Lobachevsky plane is an R-tree.

Multiple nonisometric asymptotic spaces exist, including high-cardinality variants.

Abstract

The notion of asymptotic space for an unbounded metric space has been introduced by Micha Gromov in 1980s. It is intended to capture the structure of a metric space at infinity. The most comprehensive definition of asymptotic space is given in the lahguage of Nonstandard Analysis (NSA). It turns out that the asymptotic space depends on the underlying nonstandard extension of the standard universe. This paper contains the exhaustive description of asymptotic spaces of the Lobachevski plane which turns ourt to be an R-tree. However, there turn out to be a plenty of different nonisometric asymptotic spaces, including the spaces of high cardinality.