  The group of automorphisms of a symbolic dynamical system is countable, but
often very large. For example, for a mixing subshift of finite type, the
automorphism group contains isomorphic copies of the free group on two
generators and the direct sum of countably many copies of $\mathbb{Z}$. In
contrast, the group of automorphisms of a symbolic system of zero entropy seems
to be highly constrained. Our main result is that the automorphism group of any
minimal subshift of stretched exponential growth with exponent $<1/2$, is
amenable (as a countable discrete group). For shifts of polynomial growth, we
further show that any finitely generated, torsion free subgroup of Aut(X) is
virtually nilpotent.
