  Given any connected compact orientable surface, a pair of mapping classes are
said to be procongruently conjugate if they induce a conjugate pair of outer
automophisms on the profinite completion of the fundamental group of the
surface. For example, this occurs if they induce conjugate outer automorphisms
on every characteristic finite quotient of the fundamental group. In this
paper, it is shown that every procongruent conjugacy class of mapping classes,
as a subset of the surface mapping class group, is the disjoint union of at
most finitely many conjugacy classes of mapping classes. For any pseudo-Anosov
mapping class of a connected closed orientable surface, several topological
features are confirmed to depend only on the procongurent conjugacy class of
the mapping class, including: the stretching factor, the topological type of
the prong singularities, the transverse orientability of the invariant
foliations, and the isomorphism type of the symplectic Floer homology.
