About embedded quarters and points at infinity in the hyperbolic plane

TL;DR

This paper investigates the behavior of embedded quarters in the hyperbolic plane, showing sequences converge to ends and proving the impossibility of algorithmically distinguishing these ends.

Contribution

It introduces a family of embedded quarters with converging sequences and establishes the non-existence of an algorithm to determine end equality.

Findings

Sequences of embedded quarters converge to hyperbolic plane ends.

No algorithm can decide if two ends are equal.

The results deepen understanding of hyperbolic plane topology.

Abstract

In this paper, we prove two results. First, there is a family of sequences of embedded quarters of the hyperbolic plane such that any sequence converges to a limit which is an end of the hyperbolic plane. Second, there is no algorithm which would allow us to check whether two given ends are equal or not.