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Surjunctivity does not characterize cosoficity of invariant random subgroups

Lewis Bowen, Michael Chapman

Abstract

A group is surjunctive if every injective cellular automaton on it is also surjective. Gottschalk famously conjectured that all groups are surjunctive. This remains a central open problem in symbolic dynamics and descriptive set theory. Gromov and Weiss termed the notion of sofic groups, and proved that all such groups are surjunctive, providing the largest class of groups which satisfy Gottschalk's conjecture. It is still open to decide whether all groups are sofic. This became a major open problem in group theory, and is related to other well known problems such as the Aldous--Lyons conjecture in probability theory and to Connes' embedding problem in the theory of operator algebras. A complementary natural question to ask is: Does the reverse implication to Gromov and Weiss' result holds? Namely, are all surjunctive groups sofic? As currently there are no known non-sofic groups, answering this problem in the negative in the category of groups is still out of reach. This paper resolves this problem in the generalized setup of invariant random subgroups of free groups (IRSs), where non (co)sofic objects were recently shown to exist by Lubotzky, Vidick and the two authors. Specifically, we prove that there exists a surjunctive non (co)sofic IRS, resolving the aforementioned problem in the negative. Our proof uses a complexity theoretic approach, and in particular a recent development due to Manzoor, as well as the theory of Rokhlin entropy developed by Seward and others. As a byproduct of our proof technique, the non (co)sofic IRS we provide satisfies a condition stronger than surjunctivity; it satisfies a version of Seward's maximal Rokhlin entropy of Bernoulli Shifts (RBS) criterion.

Surjunctivity does not characterize cosoficity of invariant random subgroups

Abstract

A group is surjunctive if every injective cellular automaton on it is also surjective. Gottschalk famously conjectured that all groups are surjunctive. This remains a central open problem in symbolic dynamics and descriptive set theory. Gromov and Weiss termed the notion of sofic groups, and proved that all such groups are surjunctive, providing the largest class of groups which satisfy Gottschalk's conjecture. It is still open to decide whether all groups are sofic. This became a major open problem in group theory, and is related to other well known problems such as the Aldous--Lyons conjecture in probability theory and to Connes' embedding problem in the theory of operator algebras. A complementary natural question to ask is: Does the reverse implication to Gromov and Weiss' result holds? Namely, are all surjunctive groups sofic? As currently there are no known non-sofic groups, answering this problem in the negative in the category of groups is still out of reach. This paper resolves this problem in the generalized setup of invariant random subgroups of free groups (IRSs), where non (co)sofic objects were recently shown to exist by Lubotzky, Vidick and the two authors. Specifically, we prove that there exists a surjunctive non (co)sofic IRS, resolving the aforementioned problem in the negative. Our proof uses a complexity theoretic approach, and in particular a recent development due to Manzoor, as well as the theory of Rokhlin entropy developed by Seward and others. As a byproduct of our proof technique, the non (co)sofic IRS we provide satisfies a condition stronger than surjunctivity; it satisfies a version of Seward's maximal Rokhlin entropy of Bernoulli Shifts (RBS) criterion.

Paper Structure

This paper contains 15 sections, 6 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cosoficity and Surjunctivity of Invariant Random Subgroups
  3. Proof technique
  4. Subgroup Tests and non cosofic IRSs
  5. Manzoor's amplification of the complexity theoretic framework
  6. Seward's Rokhlin entropy criterion for surjunctivity
  7. Structure of the paper
  8. Entropy Theory
  9. Basic Notions
  10. Observables and Partitions
  11. Generating observables and Rokhlin entropy
  12. The maximal Rokhlin entropy of Bernoulli Shifts criterion for Invariant Random Subgroups
  13. A condition equivalent to RBS
  14. Outer approximation
  15. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.2

There exists a surjunctive non cosofic IRS of a finitely generated free group.

Figures (2)

  • Figure 1.1: Outer and inner algorithmic approximations, together with an incomputability result, imply that the two sets are different.
  • Figure 1.2: Outer algorithmic approximation for any set of IRSs which satisfy a certain property $\mathcal{P}$, together with the incomputability result for ${\rm val}_{\rm sof}(\mathcal{T})$ from Fact \ref{['fact:sof_val_is_RE_hard']}, imply that there are non cosofic IRSs with this property $\mathcal{P}$.

Theorems & Definitions (37)

  • Theorem 1.2
  • Definition 1.3: Invariant Random Subgroups
  • Definition 1.4: Soficity and Cosoficity
  • Definition 1.5: Cellular automaton on an IRS
  • Definition 1.6: Surjunctive IRS
  • Remark 1.7
  • Remark 1.11
  • Remark 1.12
  • Definition 2.2: Generating observable
  • Remark 2.3
  • ...and 27 more