Bernoulli factors of group shifts

Definitions and mathematical statement

Let $G$ be a finite group, $d\ge1$, and $X\le G^{\mathbb Z^d}$ a closed subgroup invariant under all shifts $(\sigma^v x)_u=x_{u+v}$. Equip $X$ with the product topology and normalized Haar measure. This is a group shift; no commutativity assumption is imposed on $G$ or $X$. A Bernoulli group shift is $H^{\mathbb Z^d}$ with its shift action, where $H$ is a finite group.

An algebraic factor map is a continuous surjective group homomorphism $\pi:X\to H^{\mathbb Z^d}$ satisfying $\pi\sigma^v=\sigma^v\pi$ for every $v$. The substantive source question asks whether every nonabelian group shift has such a factor of equal entropy: \[ h_{\rm top}(X)=\log|H|. \] Here entropy is the asymptotic logarithmic growth of allowed patterns per lattice site, computed on increasingly large cubes. The full group shift on $H$ has entropy $\log|H|$.

The equal-entropy requirement is supplied by the original Boyle–Schraudner results underlying the survey. Without it the map to the one-element group would make the literal existence question vacuous. Their theorem already gives an equal-entropy topological Bernoulli factor for every group shift, and an algebraic such factor in the abelian case. The target is therefore to preserve multiplication in the nonabelian case. Neither a merely measurable factor nor a continuous factor that fails to be a homomorphism meets that requirement.

Short English statement

Does every nonabelian group shift have an equal-entropy Bernoulli factor through a continuous group homomorphism?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600025 (AMR-045-0025), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boyle - Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 17.1. Original source indexed by AMR-045-0025. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.
• S3 Boyle and Schraudner, Z^d group shifts and Bernoulli factors (2008), introductory statements and Theorem3.2. https://math.umd.edu/~mboyle/papers/gsub27aug2007.pdf.