The sails of the \(\text{SL}(2,\mathbb Z)\) operators and their symmetries (Q1035286)

The \textit{geometric continued fraction} associated to a pair of lines through the origin of irrational slope is a bi-infinite sequence of integers that reflects the geometry of the convex hull of points of integer coordinates lying (appropriately) between them. These convex hulls are the \textit{sails} of the title of this paper. The eigendirections of any hyperbolic element of \(\text{SL}(2, \mathbb Z)\) define lines of the appropriate type here; the resulting geometric continued fraction is known to be periodic. The author shows that given a period, there is a finite number of hyperbolic elements with positive entries realizing this period. Symmetries are studied in this setting.