  The set of automorphisms of a one-dimensional \shift $(X, \sigma)$ forms a
countable, but often very complicated, group. For zero entropy shifts, it has
recently been shown that the automorphism group is more tame. We provide the
first examples of countable groups that cannot embed into the automorphism
group of any zero entropy \shiftno. In particular, we show that the
Baumslag-Solitar groups ${\rm BS}(1,n)$ and all other groups that contain
exponentially distorted elements cannot embed into ${\rm Aut}(X)$ when $h_{{\rm
top}}(X)=0$. We further show that distortion in nilpotent groups gives a
nontrivial obstruction to embedding such a group in any low complexity shift.
