The Non-Euclidean Euclidean Algorithm

TL;DR

This paper presents a novel interpretation of a geometrically motivated algorithm for determining the discreteness of two-generator real Mobius groups as a non-Euclidean Euclidean algorithm, linking hyperbolic geometry with classical division algorithms.

Contribution

It introduces a new perspective by interpreting the group discreteness algorithm as a non-Euclidean Euclidean algorithm, and applies it to find shortest curves on quotient surfaces.

Findings

The algorithm determines whether a two-generator real Mobius group is discrete.

It provides a method to find the three shortest curves on the quotient surface.

The interpretation connects hyperbolic geometry with Euclidean division algorithms.

Abstract

In this paper we demonstrate how the geometrically motivated algorithm to determine whether a two generator real Mobius group acting on the Poincare plane is or is not discrete can be interpreted as a non-Euclidean Euclidean algorithm. That is, the algorithm can be viewed as an application of the Euclidean division algorithm to real numbers that represent hyperbolic distances. In the case that the group is discrete and free, the algorithmic procedure also gives a non-Euclidean Euclidean algorithm to find the three shortest curves on the corresponding quotient surface.