The fact that the euclidean algorithm eventually terminates is pervasive in
mathematics. In the language of continued fractions, it can be stated by saying
that the orbits of rational points under the Gauss map x-->1/x eventually
reach zero. Analogues of this fact for Gauss maps defined over quadratic number
fields have relevance in the theory of flows on translation surfaces, and have
been established via powerful machinery, ultimately relying on the Veech
dichotomy. In this paper, for each commensurability class of noncocompact
triangle groups of quadratic invariant trace field, we construct a Gauss map
whose defining matrices generate a group in the class; we then provide a direct
and self-contained proof of termination. As a byproduct, we provide a new proof
of the fact that noncocompact triangle groups of quadratic invariant trace
field have the projective line over that field as the set of cross-ratios of
cusps.
Our proof is based on an analysis of the action of nonnegative matrices with
quadratic integer entries on the Weil height of points. As a consequence of the
analysis, we show that long symbolic sequences in the alphabet of our maps can
be effectively split into blocks of predetermined shape having the property
that the height of points which obey the sequence and belong to the base field
decreases strictly at each block end. Since the height cannot decrease
infinitely, the termination property follows.
