The non-Euclidean Euclidean algorithm (Q2445379)

The paper draws attention to the connection between the well-known algorithm for checking discreteness of a two-generator Fuchsian group [\textit{J. Gilman} and \textit{B. Maskit}, Mich. Math. J. 38, No. 1, 13--32 (1991; Zbl 0724.20033)] and the Euclidean division algorithm. As a byproduct, it is shown that, in the case when the group is discrete and free, the algorithm also provides the lengths of the three shortest curves on the corresponding quotient surface.