The non-Euclidean Euclidean algorithm

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.