The non-Euclidean Euclidean algorithm

Jane Gilman

doi:https://doi.org/10.1016/j.aim.2013.09.012

The non-Euclidean Euclidean algorithm

Jane Gilman

Rutgers Newark, Math & Computer Science

Rutgers, The State University

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

In this paper we demonstrate how the geometrically motivated algorithm to determine whether a two-generator real Möbius group acting on the Poincaré 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.

Original language	American English
Pages (from-to)	227-241
Number of pages	15
Journal	Advances in Mathematics
Volume	250
DOIs	https://doi.org/10.1016/j.aim.2013.09.012
State	Published - Jan 15 2014

ASJC Scopus subject areas

Mathematics(all)

Keywords

Algorithms
Discreteness criteria
Hyperbolic geometry
Kleinian groups
Teichmuller theory

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https://doi.org/10.1016/j.aim.2013.09.012

Cite this

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title = "The non-Euclidean Euclidean algorithm",

abstract = "In this paper we demonstrate how the geometrically motivated algorithm to determine whether a two-generator real M{\"o}bius group acting on the Poincar{\'e} 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.",

keywords = "Algorithms, Discreteness criteria, Hyperbolic geometry, Kleinian groups, Teichmuller theory",

author = "Jane Gilman",

note = "Funding Information: The author would like to thank Bill Goldman, Ryan Hoban, and Sergei Novikov for some useful and thought provoking conversations and e-mail exchanges. Thanks are also due to the Mathematics Department at the CUNY Graduate Center for its hospitality during the authorʼs sabbatical. This work was partially supported by the NSF when the author was on an IPA appointment which included released time for research. The NSF has no responsibility for the content.",

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N2 - In this paper we demonstrate how the geometrically motivated algorithm to determine whether a two-generator real Möbius group acting on the Poincaré 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.

AB - In this paper we demonstrate how the geometrically motivated algorithm to determine whether a two-generator real Möbius group acting on the Poincaré 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.

KW - Algorithms

KW - Discreteness criteria

KW - Hyperbolic geometry

KW - Kleinian groups

KW - Teichmuller theory

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JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

The non-Euclidean Euclidean algorithm

Abstract

ASJC Scopus subject areas

Keywords

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