  This paper initiates a systematic study of the relation of commensurability
of surface automorphisms, or equivalently, fibered commensurability of
3-manifolds fibering over the circle. We show that every hyperbolic fibered
commensurability class contains a unique minimal element, whereas the class of
Seifert manifolds fibering over the circle consists of a single
commensurability class with infinitely many minimal elements. The situation for
non-geometric manifolds is more complicated, and we illustrate a range of
phenomena that can occur in this context.
