Surjunctivity does not characterize cosoficity of invariant random subgroups

TL;DR

This paper constructs a counterexample in the setting of invariant random subgroups of free groups, showing that surjunctivity does not imply soficity, thus resolving a key open problem in group theory and symbolic dynamics.

Contribution

It demonstrates the existence of a surjunctive but non (co)sofic invariant random subgroup, using a complexity theoretic approach and Rokhlin entropy theory.

Findings

Existence of a surjunctive non (co)sofic IRS.

Counterexample resolves the implication question in the generalized IRS setting.

The IRS satisfies a stronger condition than surjunctivity, related to Rokhlin entropy.

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.