The Automorphism Group of a Minimal Shift of Stretched Exponential Growth

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 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. 1. Complexity and the automorphism group Let (X, σ) be a subshift over the finite alphabet A, meaning that X ⊂ A is closed and invariant under the left shift σ : A → A. The group of automorphisms Aut(X) of (X, σ) is the group of homeomorphisms of φ : X → X such that φ ◦ σ = σ ◦ φ. A classic result of Curtis, Hedlund, and Lyndon is that Aut(X) is always countable, but a number of results have shown that Aut(X) can be quite large. For example, for any mixing subshift of finite type, Aut(X) always contains (among others) a copy of: every finite group, the direct sum of countably many copies of Z, and the free group on two generators [11, 2]; every countable, locally finite, residually finite group [13]; the fundamental group of any 2-manifold [13]. This extremely rich subgroup structure makes the problem of deciding when two shifts have isomorphic automorphism groups challenging. Moreover, Kim and Roush [13] showed that the automorphism group of any full shift is contained in the automorphism group of any other full shift (and more generally is contained in any mixing subshift of finite type), thereby dooming any strategy for distinguishing two such groups that relies on finding a subgroup of one that does not embed into the other. Even the question of whether the automorphism groups of the full 2-shift and the full 3-shift are isomorphic remains a difficult open problem [2] (although, as they remark, the automorphism groups of the full 2-shift and the full 4-shift are not isomorphic). In all of these examples, the complicated nature of Aut(X) is a manifestation of the relatively light constraints required on x ∈ A to be a member of the shift space X . Another consequence of this fact is that these shifts always have positive (although arbitrarily small) entropy. As a corollary, if G is a group that embeds into the automorphism group of the full 2-shift, then for any h > 0 there is a subshift of topological entropy less than h into whose automorphism group G also embeds. 2010 Mathematics Subject Classification. 37B10, 43A07, 68R15. The second author was partially supported by NSF grant 1500670. 1 2 VAN CYR AND BRYNA KRA It is therefore natural to ask whether the automorphism group of a zero entropy subshift is more highly constrained than its positive entropy relatives. Over the past several years, the authors [4, 5] and others (e.g., [3, 7, 14, 15]) have shown that the zero entropy case is indeed significantly more constrained, and we continue this theme in the present work. Specifically, we study how the the growth rate of the factor complexity PX(n), the number of nonempty cylinder sets of length n, constrains the algebraic properties of group of automorphisms. For a shift whose factor complexity grows at most linearly, we showed [5] that every finitely generated subgroup of Aut(X) is virtually Z for some d that depends on the growth rate. We further showed [4] that for a transitive shift of subquadratic growth, the quotient of Aut(X) by the subgroup generated by σ, is periodic. Further examples of minimal shifts with polynomial complexity and highly constrained automorphism groups were constructed by Donoso, Durand, Maass and Petite [7], and an example of a minimal shift with subquadratic growth whose automorphism group is not finitely generated was given by Salo [14]. Our main theorem provides a strong constraint on Aut(X) for any minimal subshift of stretched exponential growth with exponent < 1/2. We show: Theorem 1.1. If (X, σ) is a minimal shift such that there exists β < 1/2 satisfying lim sup n→∞ log(PX(n)) nβ = 0, (1) then Aut(X) is amenable. Moreover, every finitely generated, torsion free subgroup of Aut(X) has subexponential growth. For minimal shifts of polynomial growth, we show more: Theorem 1.2. If (X, σ) is a minimal shift such that there exists d ∈ N satisfying lim sup n→∞ PX(n) nd = 0, (2) then Aut(X) is amenable. Furthermore, every finitely generated, torsion free subgroup of Aut(X) is virtually nilpotent with polynomial growth rate at most d − 1. In particular, the step of the nilpotent subgroup is at most ⌊