The Automorphism Group of a Shift of Slow Growth Is Amenable

Suppose (X,σ) is a subshift, PX(n) is the word complexity function of X, and Aut(X) is the group of automorphisms of X. We show that if PX(n) = o(n 2/ log n), then Aut(X) is amenable (as a countable, discrete group). We further show that if PX(n) = o(n 2), then Aut(X) can never contain a nonabelian free semigroup (and, in particular, can never contain a nonabelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a semigroup. 1. Amenability and the automorphism group For a subshift (X,σ) over a finite alphabet, let Aut(X) = Aut(X,σ) denote the group of all automorphisms of the system, meaning the collection of all homeomorphisms φ : X → X such that φ ◦ σ = σ ◦ φ. The automorphism group of many subshifts with positive entropy, including the full shift and more generally any mixing shift of finite type, is a countable group that contains many structures, including isomorphic copies of any finite group, countably many copies of Z, and the free group on countably many generators (see [7, 1]). In particular, when given the discrete topology, these automorphism groups are never amenable. This behavior is in contrast to what happens in minimal shifts of zero entropy: if the complexity function PX(n), which counts the number of words in the language of the shift, satisfies lim supn→∞ log(PX(n)) nβ = 0 for some β < 1/2, then the automorphism group Aut(X) is amenable; furthermore, every finitely generated torsion-free subgroup of the automorphism group has subexponential growth [4]. For lower complexities, one can sometimes carry out a more detailed analysis of the automorphism group, and this is done for polynomial growth in [4], and with extra assumptions on the dynamics, sometimes one can give a complete description of the automorphism group (see [2, 3, 5]). We continue the systematic study of automorphism groups here, focusing on subshifts with zero entropy. These automorphism groups are constrained by the subexponential growth rate of words in the language of the shift, and it seems plausible that for any subshift (X,σ) of zero entropy, we have a version of the Tits alternative: either Aut(X) contains a free subgroup or Aut(X) is amenable. It may be possible that a stronger alternative holds, namely either Aut(X) contains a free subgroup or it is virtually abelian. Somewhat surprisingly, we can not rule out that such an alternative holds for any shift, even without an assumption on the entropy. 2010 Mathematics Subject Classification. 37B50 (primary), 43A07, 68R15.