Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A geometric obstruction to self-simulation for groups

Sebastián Barbieri, Kanéda Blot, Mathieu Sablik, Ville Salo

TL;DR

The paper introduces a new quasi-isometry invariant, the extraterrestrial property, defined via $(m,k,r)$-UFOs in Cayley graphs, and proves that any finitely generated extraterrestrial group carries an effectively closed-by-pattern subshift that is not a factor of any group-SFT, providing a robust obstruction to self-simulation. Central to the work is a geometric generalization of the mirror shift, the generalized mirror shift $X_{ exttt{GM}}$, which is effective but not sofic, forcing a contradiction if it were to arise as a factor of a $ ext{G}$-SFT. The authors develop tools showing extraterrestriality is preserved under quasi-isometries and extend the property to broad classes of groups, including amenable, multi-ended, amalgamated/HNN constructions, generalized Baumslag–Solitar groups, cocompact Fuchsian groups, and fundamental groups of surfaces. These results systematically rule out self-simulability for wide families of groups and provide a geometric route to identifying non-sofic effective subshifts, with implications for the computability of group actions and symbolic dynamics on groups.

Abstract

We introduce a new quasi-isometry invariant for finitely generated groups and show that every group with this property admits a subshift which is effectively closed by patterns and that cannot be realized as the topological factor of any subshift of finite type. We provide several examples of groups with the property, such as amenable groups, multi-ended groups, generalized Baumslag-Solitar groups, fundamental groups of surfaces, and cocompact Fuchsian groups.

A geometric obstruction to self-simulation for groups

TL;DR

The paper introduces a new quasi-isometry invariant, the extraterrestrial property, defined via -UFOs in Cayley graphs, and proves that any finitely generated extraterrestrial group carries an effectively closed-by-pattern subshift that is not a factor of any group-SFT, providing a robust obstruction to self-simulation. Central to the work is a geometric generalization of the mirror shift, the generalized mirror shift , which is effective but not sofic, forcing a contradiction if it were to arise as a factor of a -SFT. The authors develop tools showing extraterrestriality is preserved under quasi-isometries and extend the property to broad classes of groups, including amenable, multi-ended, amalgamated/HNN constructions, generalized Baumslag–Solitar groups, cocompact Fuchsian groups, and fundamental groups of surfaces. These results systematically rule out self-simulability for wide families of groups and provide a geometric route to identifying non-sofic effective subshifts, with implications for the computability of group actions and symbolic dynamics on groups.

Abstract

We introduce a new quasi-isometry invariant for finitely generated groups and show that every group with this property admits a subshift which is effectively closed by patterns and that cannot be realized as the topological factor of any subshift of finite type. We provide several examples of groups with the property, such as amenable groups, multi-ended groups, generalized Baumslag-Solitar groups, fundamental groups of surfaces, and cocompact Fuchsian groups.

Paper Structure

This paper contains 14 sections, 24 theorems, 24 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shift spaces
  4. Computability of group actions
  5. UFOs and extraterrestrial graphs
  6. Extraterrestrial groups are not self-simulable
  7. The generalized mirror shift
  8. Proof of \ref{['mainthm:property_ET_groups_not_SS']}
  9. Examples of extraterrestrial groups
  10. Amenable and multi-ended groups
  11. Amenable extensions of multi-ended graphs
  12. Amalgamated free products and HNN extensions
  13. Groups quasi-isometric to the hyperbolic plane
  14. An amenable cut in the surface group

Key Result

Theorem A

Being extraterrestrial is a quasi-isometry invariant for graphs of bounded degree.

Figures (5)

  • Figure 1.1: A portion of a configuration in the $\mathbb{Z}^2$-mirror shift.
  • Figure 1.2: A $(1,4,18)$-UFO for the canonical Cayley graph of $\mathbb{Z}^2$.
  • Figure 5.1: A finite portion of the Bass-Serre tree of $\mathrm{BS}(1,2)$
  • Figure 5.2: A finite portion of the pentagon model.
  • Figure 5.3: The cut $X$ used in \ref{['thm:SurfaceCut']}.

Theorems & Definitions (59)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • ...and 49 more