Soficity of free extensions of effective subshifts

TL;DR

The paper establishes a dichotomy for free extensions of effectively closed subshifts over direct products $G=H\times K$. When $K$ is nonamenable and $H$ has decidable word problem, every effectively closed $H$-subshift has a sofic free extension to $G$, proven via a novel rooted-SFT construction that embeds cosets into pairwise disjoint binary trees and simulates a Turing machine with tentacle communication. Conversely, when $H$ and $K$ are both amenable, there exist effectively closed subshifts whose free extensions are not sofic, demonstrated through reflection and ball-mimic type shifts. These results yield a new simulation theorem and construct strongly aperiodic SFTs on certain direct products, advancing the computability-dynamics interface for group actions of nonamenable groups. Collectively, the work broadens the landscape of soficity for group shifts and connects paradoxical decompositions with explicit SFT constructions.

Abstract

Let $G$ be a group and $H\leqslant G$ a subgroup. The free extension of an $H$-subshift $X$ to $G$ is the $G$-subshift $\widetilde{X}$ whose configurations are those for which the restriction to every coset of $H$ is a configuration from $X$. We study the case of $G = H \times K$ for infinite and finitely generated groups $H$ and $K$: on the one hand we show that if $K$ is nonamenable and $H$ has decidable word problem, then the free extension to $G$ of any $H$-subshift which is effectively closed is a sofic $G$-subshift. On the other hand we prove that if both $H$ and $K$ are amenable, there are always $H$-subshifts which are effectively closed by patterns whose free extension to $G$ is non-sofic. We also present a few applications in the form of a new simulation theorem and a new class of groups which admit strongly aperiodic SFTs.

Figures (6)

• Figure 1: Part of a configuration of the reflection shift in $\mathbb Z^2$.
• Figure 2: The terniary structure induced by $\mathbf{T}$. The three incoming arrows represent $s_1(a),s_2(a),s_3(a)$, the outgoing arrow represents $p(a)$ and the number $c(a)$. We mark in red the binary tree associated to the topmost element according to $\gamma$. We can see that the choice in the definition of $\gamma$ forces the binary trees to be pairwise disjoint.
• Figure 3: The computation branches in $\mathbb Z \times \{0,1\}^*$. Each computation zone is of the form $\blacktriangleright(\blacksquare)^*\blacktriangleleft$. The $\texttt{S}_i$ symbols are seeds and the $\varnothing$ symbols are placeholders for unused space. The numbers indicate the branch in which the computation will continue, note that the numbers are constant inside every computation zone.
• Figure 4: The alphabet $A^{0}_{\mathcal{T}}$ associated to a Turing machine $\mathcal{T} = \{Q,\Sigma, \delta,q_0,q_F\}$. $\sqcup$ is the blank symbol, $q_0$ the starting state. For the bottom row tiles, $a \in \Sigma$ is an arbitrary symbol and $q \in Q$ is an arbitrary state. $(s,b),(\ell,c),(r,d) \in Q \times \Sigma$ are pairs such that $\delta(s,b)=(s',b',0)$, $\delta(\ell,c)=(\ell',c',-1)$ and $\delta(r,d)=(r',d',+1)$.
• Figure 5: An example of a tiling of the computation layer. Note that fixing the seed symbol on the bottom position determines completely the rest of the tiling. Here we illustrate a Turing machine with alphabet $\Sigma = \{\sqcup,a,b\}$, $Q = \{q_0,r,s,t\}$ and with $\delta(q_0,\sqcup) =(a,r,1)$, $\delta(r,\sqcup) = (s,b,0)$, $\delta(s,b) = (a,t,-1)$, $\delta(t,a) = (b,s,1)$.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6: Curtis-Lyndon-Hedlund Hedlund1969