The non-Euclidean Euclidean algorithm

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1 Scopus citations


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 languageAmerican English
Pages (from-to)227-241
Number of pages15
JournalAdvances in Mathematics
StatePublished - Jan 15 2014

ASJC Scopus subject areas

  • Mathematics(all)


  • Algorithms
  • Discreteness criteria
  • Hyperbolic geometry
  • Kleinian groups
  • Teichmuller theory


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