  We study the word length entropy of automorphisms of residually nilpotent
groups, and how the entropy of such group automorphisms relates to the entropy
of induced automorphisms on various nilpotent quotients. We show that much like
the structure of a nilpotent group is dictated to a large degree by its
abelianization, the entropy of an automorphism of a nilpotent group is dictated
by its entropy on the abelianization. We give some applications to the study of
pseudo-Anosov homeomorphisms of surfaces. In particular, we show that if $\psi$
is a non--homological pseudo-Anosov homeomorphism of a surface $\Sigma$ with
dilatation $K$ and $N$ is any nilpotent quotient of any finite index
characteristic subgroup of $\pi_1(\Sigma)$ to which $\psi$ descends, the
entropy of $\psi$ viewed as an automorphism of $N$ is bounded away from $K$.
This answers a question of D. Sullivan.
