The Non-Euclidean Euclidean Algorithm
Group Theory
2013-09-27 v4 Complex Variables
Geometric Topology
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.
Cite
@article{arxiv.1207.1062,
title = {The Non-Euclidean Euclidean Algorithm},
author = {Jane Gilman},
journal= {arXiv preprint arXiv:1207.1062},
year = {2013}
}
Comments
To appear, Advances in Mathematics 16 pages, 3 figures; typo on line 7 of theorem 3.1 of version from day earlier fixed