  A basic result in the elementary theory of continued fractions says that two
real numbers share the same tail in their continued fraction expansions iff
they belong to the same orbit under the projective action of PGL(2,Z). This
result was first formulated in Serret's Cours d'alg\`ebre sup\'erieure, so
we'll refer to it as to 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 subgroups
Sigma of PGL(2,Z) generated by the branches of the Gauss maps in a large family
of algorithms, and that each Sigma-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.
