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# Jónsson-Tarski algebraic subshifts

A big theme in my research is looking at internal algebras in the category of all subshifts and block-maps between them. In this post, we study subshifts in a not-very-well-known variety and prove a little theorem.

Here, a subshift is a shift-invariant closed subset of $A^G$ where $A$ is a finite alphabet (discrete set) and $G$ is some discrete group (which stays fixed during the discussion). It is a dynamical system under the shift-action of $G$. The category of subshifts has subshifts as objects, and shift-commuting continuous functions between them as morphisms.

Concretely an internal algebra in this category just means that the set of elements of the algebra is a subshift, the operations are given by shift-commuting continuous function, and the algebra axioms hold in the usual set-theoretic sense.

A Jónsson-Tarski algebra is an algebra with a binary operation $w$ and unary operations $p_1, p_2$, with identities $p_1(w(a, b)) = a$, $p_2(w(a, b)) = b, w(p_1(a), p_2(a)) = a$.

The axioms essentially say that if $A$ is a Jónsson-Tarski algebra, then $w$ is an isomorphism from $A \times A$ to $A$. In particular, a Jónsson-Tarski algebra is either empty, a singleton, or infinite. Conversely, in the category Set, in classical mathematics, any infinite set has the structure of a Jónsson-Tarski algebra.

It is a theme in symbolic dynamics to try to understand various types of finiteness in the category of subshifts (e.g. surjunctivity, amenability and soficity are finiteness properties of groups that have links to cellular automata). Thus, it is of interest to try to understand whether a subshift can be infinite in the Jónsson-Tarski sense, i.e. whether the category of subshifts has internal Jónsson-Tarski algebras.

Another reason why Jónsson-Tarski algebras are interesting from a symbolic dynamical point of view is that the automorphism group of a Jónsson-Tarski algebraic object contains a copy of Thompson’s $V$. It is known that Thompson’s $V$ embeds in the automorphism group of a one-dimensional SFT, so it is an interesting question whether one can even have a “natural” copy of it.

We show that there are no Jónsson-Tarski algebraic subshifts when the spatially acting group $G$ has subexponential growth, i.e. when the size of balls grows at subexponential rate. This covers e.g. virtually nilpotent groups and the Grigorchuk group.

Theorem. Let $G$ be a finitely-generated group with subexponential growth, and let $X \subset A^G$ be a Jónsson-Tarski algebraic subshift. Then $X$ has at most one point.

Proof. The map $w : X \times X \to X$ is a topological conjugacy between the $G$-subshift $Y = X \times X$ (under the diagonal action; we can see it as a subshift over alphabet $B = A^2$) and $X$, with inverse $p = p_1 \times p_2 : X \to X \times X$. These maps are shift-commuting and continuous, thus they are block maps.

Let us recall what this means. Fix a shift-convention, say the action of $G$ is defined by the formula $g \cdot x_h = x_{g^{-1} h}$. Then $f : X \to Y$ is a block map if there exists $r \geq 0$ such that $f(x)_g$ is determined by the contents of $x|_{g B_r}$, where $B_r$ is the ball of radius $r$ with respect to some finite generating set, by some uniform local rule. More precisely, there exists $F : A^{B_r} \to B$ such that $f(x)_g = F(P)$ where $P(h) = x_{g h}$.

Now if $r$ is a radius for our operation $p : X \to Y$, then for $x \in X$, $p$ determines from the contents $x_{B_{R+r}}$ the contents of $p(x)_{B_R}$.

Let $Y = X \times X$, and observe that in particular since $p$ is a conjugacy, it is surjective. Thus, for a subshift $Z$ writing $c_n(Z) = |{z_{B_n} \;|\; z \in Z}|$, we must have $c_{R+r}(X) \geq c_R(Y) = c_R(X)^2$.

In particular $c_{R+kr}(X) \geq c_R(X)^{2^k}$. But $c_{R+kr}(X) \leq |A|^{B_{R+kr}}$ (this is how many patterns there are in the full shift $A^G$), so $|A|^{|B_{R+kr}}| \geq c_R(X)^{2^k}$, from which we solve $|B_{R+kr}| \geq 2^k \log_2 c_R(X) / \log_2(|A|)$ and get that the group has exponential growth, as soon as $c_R(X) \geq 2$ for any radius $R$. Square.

The proof does not work for general amenable groups, let alone a general group. I don’t know if there is a smarter version of the proof that uses Følner sequences in place of balls somehow. On the other hand I don’t have an example of a Jónsson Tarski algebraic subshift on any group.