Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Around Gromov's injectivity lemma and applications to post-injunctive groups

Xuan Kien Phung

Abstract

Gottschalk's surjunctivity conjecture states that for all group universes and finite alphabets, every equivariant and continuous selfmap of the full shift, known as cellular automaton, cannot be a strict embedding. Not all surjective cellular automata are injective. However, if the surjectivity condition is replaced by a certain strengthened property called post-surjectivity then all post-surjective cellular automata must be bijective whenever the universe is a sofic group. A group universe is said to be post-injunctive if every post-surjective cellular automaton with finite alphabet over this group universe must be bijective. Gromov's injectivity lemma states each injective cellular automaton over a subshift can be extended to an injective cellular automaton over every subshift which is close enough to the initial subshift. In this paper, we obtain analogous results where injectivity is replaced by other fundamental dynamical properties namely post-surjectivity and pre-injectivity. We also study various stable properties of the class of post-injunctive groups in parallel to properties of surjunctive groups. Among the results, we show that semidirect extensions of post-injunctive groups with residually finite kernels must be post-injunctive.

Around Gromov's injectivity lemma and applications to post-injunctive groups

Abstract

Gottschalk's surjunctivity conjecture states that for all group universes and finite alphabets, every equivariant and continuous selfmap of the full shift, known as cellular automaton, cannot be a strict embedding. Not all surjective cellular automata are injective. However, if the surjectivity condition is replaced by a certain strengthened property called post-surjectivity then all post-surjective cellular automata must be bijective whenever the universe is a sofic group. A group universe is said to be post-injunctive if every post-surjective cellular automaton with finite alphabet over this group universe must be bijective. Gromov's injectivity lemma states each injective cellular automaton over a subshift can be extended to an injective cellular automaton over every subshift which is close enough to the initial subshift. In this paper, we obtain analogous results where injectivity is replaced by other fundamental dynamical properties namely post-surjectivity and pre-injectivity. We also study various stable properties of the class of post-injunctive groups in parallel to properties of surjunctive groups. Among the results, we show that semidirect extensions of post-injunctive groups with residually finite kernels must be post-injunctive.
Paper Structure (8 sections, 12 theorems, 77 equations)

This paper contains 8 sections, 12 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Restriction and cellular automata over quotient groups
  3. The space of marked groups
  4. Post-injunctive groups
  5. On fully residually and locally post-injunctive groups
  6. Closedness of post-injunctive groups under certain semidirect extensions
  7. Around Gromov's injectivity lemma
  8. Closedness of marked post-injunctive groups

Key Result

Lemma 4.1

Let $\tau\colon A^G \to A^G$ be a cellular automaton over an alphabet $A$ and a group universe $G$. Then the following conditions are equivalent:

Theorems & Definitions (25)

  • Definition 1.1
  • Definition 1.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3: Uniform post-surjectivity
  • Lemma 5.1
  • proof
  • Corollary 5.2
  • ...and 15 more