We study the action of the mapping class group with one marked point on the
rational homology of finite nilpotent covers of a hyperbolic Riemann surface.
We use the homological representation of the mapping class to construct a
faithful infinite-dimensional representation of the mapping class group. We
show that this representation detects the Nielsen-Thurston classification of
each mapping class. We then discuss some examples that occur in the theory of
braid groups. Finally, we discuss an analogous theory for automorphisms of free
groups.