Slow continued fractions, transducers, and the Serret theorem

Notwithstanding the abundance of continued fraction algorithms in the literature, a uniform treatment of the Serret result seems missing. In this paper we show that there are finitely many possibilities for the groups Σ≤PGL2Z generated by the branches of the Gauss maps in a large family of algorithms, and that each Σ-equivalence class of reals is partitioned in finitely many tail-equivalence classes, whose number we bound. Our approach is through the finite-state transducers that relate Gauss maps to each other. They constitute opfibrations of the Schreier graphs of the groups, and their synchronizability-which may or may not hold-assures the a.e. validity of the Serret theorem.