Distortion and the Automorphism Group of a Shift

The set of automorphisms of a one-dimensional subshift (X,σ) 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 subshift. In particular, we show that the Baumslag-Solitar groups BS(1, n) and all other groups that contain exponentially distorted elements cannot embed into Aut(X) when htop(X) = 0. We further show that distortion in nilpotent groups gives a nontrivial obstruction to embedding such a group in any low complexity shift.