Equality of Rokhlin and Sofic Entropies

Definitions and mathematical statement

In the usual framework, let $G$ be a countable discrete sofic group acting on a standard probability space $(X,\mathcal B,\mu)$. The action is probability-measure-preserving if $\mu(gA)=\mu(A)$ for every $g\in G$ and $A\in\mathcal B$, free if almost every point has trivial stabilizer, and ergodic if every invariant measurable set has measure zero or one. Countability and the standard-probability-space framework are conventional hypotheses for these entropy invariants but are not explicitly stated in the supplied record.

A countable measurable partition $\alpha$ is generating if the translates $g\alpha$, for $g\in G$, generate $\mathcal B$ modulo null sets. Write \[ H_\mu(\alpha)=-\sum_{A\in\alpha}\mu(A)\log\mu(A), \qquad 0\log 0=0. \] For an ergodic action, its Rokhlin entropy is \[ h_G^{\mathrm{Rok}}(X,\mu) =\inf\{H_\mu(\alpha):\alpha\text{ is a countable generating measurable partition}\}, \] with the infimum of the empty set interpreted as $+\infty$.

A sofic approximation is a sequence $\Sigma=(\sigma_i:G\to\operatorname{Sym}(d_i))_{i\geq1}$, with $d_i\to\infty$, such that for every $g,h\in G$, \[ \frac{1}{d_i}\left|\{v:\sigma_i(gh)v=\sigma_i(g)\sigma_i(h)v\}\right|\longrightarrow1, \] and, for every distinct $g,h\in G$, \[ \frac{1}{d_i}\left|\{v:\sigma_i(g)v\ne\sigma_i(h)v\}\right|\longrightarrow1. \] Write $h_\Sigma(G\curvearrowright X,\mu)$ for the standard sofic measure entropy relative to $\Sigma$, using the same logarithm convention as for Rokhlin entropy. This invariant is defined through asymptotic counts of finite approximate models of the action and can take the value $-\infty$.

Question, with the approximation made explicit. For a free ergodic probability-measure-preserving action of a sofic group, and a fixed sofic approximation $\Sigma$, does \[ h_\Sigma(G\curvearrowright X,\mu)\ne-\infty \quad\Longrightarrow\quad h_G^{\mathrm{Rok}}(X,\mu)=h_\Sigma(G\curvearrowright X,\mu) \] hold? The hypothesis does not exclude $+\infty$, so the equality is understood in the extended real numbers. The supplied formulation does not specify a sofic approximation or quantify over approximations. The display therefore makes the necessary parameter visible without resolving whether the source intends the assertion for every approximation or for a particular one.

Short English statement

The question asks whether every free ergodic probability-measure-preserving action of a sofic group has equal Rokhlin and sofic entropies whenever the latter is not $-\infty$. Sofic entropy requires a choice of sofic approximation, whose intended quantification is unspecified. The supplied record labels the problem open; this status has not been independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $30003209$, title Equality of Rokhlin and Sofic Entropies, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year 2016 and names no individual proposer.
• Supplied report citation: Measured Group Theory (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/41, pp. 2347--2397. https://doi.org/10.4171/owr/2016/41. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Bowen, Examples in entropy theory of countable group actions. https://arxiv.org/abs/1704.06349.
• Supplied literature source: OWR Entropy report. https://publications.mfo.de/bitstream/handle/mfo/3545/OWR_2016_41.pdf?isAllowed=y&sequence=1.
• Literature assessment dated exactly 2026-08-21: the supplied curation states that Rokhlin entropy upper bounds sofic entropy and that equality for free sofic actions with sofic entropy different from $-\infty$ remains open. This is supplied dated curation, derived from the OWR material, and has not been independently verified; time-sensitive status and cited claims require independent checking.
• The registry gives statement status corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting preservation of the mathematical target. These are supplied review claims, not independent mathematical verification of this draft.