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Distortion and the automorphism group of a shift

Dedicated to the memory of Roy Adler

BK: Partially supported by NSF grant 1500670.
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  • 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 ${\rm BS}(1,n)$ 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.

    Mathematics Subject Classification: Primary: 37B10; Secondary: 22F50, 37B15, 54H20.


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