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Self-simulable groups

Sebastián Barbieri, Mathieu Sablik, Ville Salo

Abstract

We say that a finitely generated group $Γ$ is self-simulable if every effectively closed action of $Γ$ on a closed subset of $\{\texttt{0},\texttt{1}\}^{\mathbb{N}}$ is the topological factor of a $Γ$-subshift of finite type. We show that self-simulable groups exist, that any direct product of non-amenable finitely generated groups is self-simulable, that under technical conditions self-simulability is inherited from subgroups, and that the subclass of self-simulable groups is stable under commensurability and quasi-isometries of finitely presented groups. Some notable examples of self-simulable groups obtained are the direct product $F_k \times F_k$ of two free groups of rank $k \geq 2$, non-amenable finitely generated branch groups, the simple groups of Burger and Mozes, Thompson's $V$, the groups $\operatorname{GL}_n(\mathbb{Z})$, $\operatorname{SL}_n(\mathbb{Z})$, $\operatorname{Aut}(F_n)$ and $\operatorname{Out}(F_n)$ for $n \geq 5$; The braid groups $B_m$ for $m \geq 7$, and certain classes of RAAGs. We also show that Thompson's $F$ is self-simulable if and only if $F$ is non-amenable, thus giving a computability characterization of this well-known open problem. We also exhibit a few applications of self-simulability on the dynamics of these groups, notably, that every self-simulable group with decidable word problem admits a nonempty strongly aperiodic subshift of finite type.

Self-simulable groups

Abstract

We say that a finitely generated group is self-simulable if every effectively closed action of on a closed subset of is the topological factor of a -subshift of finite type. We show that self-simulable groups exist, that any direct product of non-amenable finitely generated groups is self-simulable, that under technical conditions self-simulability is inherited from subgroups, and that the subclass of self-simulable groups is stable under commensurability and quasi-isometries of finitely presented groups. Some notable examples of self-simulable groups obtained are the direct product of two free groups of rank , non-amenable finitely generated branch groups, the simple groups of Burger and Mozes, Thompson's , the groups , , and for ; The braid groups for , and certain classes of RAAGs. We also show that Thompson's is self-simulable if and only if is non-amenable, thus giving a computability characterization of this well-known open problem. We also exhibit a few applications of self-simulability on the dynamics of these groups, notably, that every self-simulable group with decidable word problem admits a nonempty strongly aperiodic subshift of finite type.

Paper Structure

This paper contains 29 sections, 43 theorems, 94 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group actions
  4. Effectively closed actions
  5. Shift spaces
  6. Effectively closed subshifts
  7. Self-simulable groups
  8. Two dynamical consequences of self-simulation
  9. Obstructions to self-simulation
  10. The paradoxical subshift
  11. The paradoxical subshift
  12. Existence of effective subshift extensions
  13. Self-simulable groups
  14. The computation tilespace
  15. Proof of Theorem \ref{['thm:selfsimulation']}
  16. ...and 14 more sections

Key Result

Theorem 1.2

The direct product $\Gamma = \Gamma_1 \times \Gamma_2$ of any pair of finitely generated non-amenable groups $\Gamma_1$ and $\Gamma_2$ is self-simulable.

Figures (4)

  • Figure 1: Part of a configuration in $X$ for $\Gamma = F_2 \times \mathbb Z$ (left) and $\Gamma =F_1 \times \mathbb Z= \mathbb Z \times \mathbb Z$ (right).
  • Figure 2: The local structure of the paradoxical subshift. The two first components of the alphabet are drawn with dashed arrows, while the third component is drawn with a thick arrow. The three conditions in the definition simply correspond to the property that following an arrow of a certain color and then following the inverse arrow of the same color amounts to no movement at all.
  • Figure 3: The alphabet $\mathcal{W}_{\mathcal{M}}$ associated to a Turing machine $\mathcal{M} = \{Q,\Sigma, q_0,q_F, \delta \}$. $a \in \Sigma$ is an arbitrary symbol and $q \in Q$ is an arbitrary state. $(s,b),(\ell,c),(r,d) \in Q \times \Sigma$ are any pairs such that $\delta(s,b)=(s',b',0)$, $\delta(\ell,c)=(\ell',c',-1)$ and $\delta(r,d)=(r',d',+1)$.
  • Figure 4: A pattern of the Wang tiling associated to the machine of Example \ref{['ex:turingo_no_machina']}. Each row, starting from the bottom, can be associated to an iteration of the Turing machine on an empty tape.

Theorems & Definitions (128)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 118 more