Distal Actions with Negative Infinite Sofic Entropy

Definitions and mathematical statement

The supplied question asks whether a distal action can have sofic topological entropy equal to $-\infty$. Its ambient group, space, and quantification over sofic approximations are not specified. The following is a precise formulation under the usual convention of countable discrete sofic groups acting on nonempty compact metrizable spaces; these conventions require confirmation against the source.

Let $G$ be a countable discrete group. A sofic approximation is a sequence $\Sigma=(\sigma_i\colon G\to\operatorname{Sym}(d_i))_{i\geq1}$, with $d_i\to\infty$, such that, for every $s,t\in G$, \[ \frac{1}{d_i}\left|\{v:\sigma_i(st)v=\sigma_i(s)\sigma_i(t)v\}\right|\longrightarrow1, \] and, for every distinct $s,t\in G$, \[ \frac{1}{d_i}\left|\{v:\sigma_i(s)v\ne\sigma_i(t)v\}\right|\longrightarrow1. \] The existence of such a sequence means that $G$ is sofic.

Let $G\curvearrowright X$ be an action by homeomorphisms on a nonempty compact metrizable space, and let $\rho$ be a compatible metric. The action is distal if \[ \forall x,y\in X,\quad x\ne y\ \Longrightarrow\ \inf_{g\in G}\rho(gx,gy)>0. \] For a nonempty finite set $F\subseteq G$ and $\delta>0$, define \[ \operatorname{Map}(\rho,F,\delta,\sigma_i) =\left\{\phi\colon\{1,\ldots,d_i\}\to X: \max_{s\in F}\left(\frac1{d_i}\sum_{v=1}^{d_i} \rho(s\phi(v),\phi(\sigma_i(s)v))^2\right)^{1/2}<\delta\right\}. \] For maps $\phi,\psi$, put $\rho_\infty(\phi,\psi)=\max_v\rho(\phi(v),\psi(v))$. Write $N_\varepsilon(Y,\rho_\infty)$ for the largest cardinality of a subset of $Y$ whose distinct points have distance at least $\varepsilon$, with $N_\varepsilon(\varnothing,\rho_\infty)=0$. In this convention, the sofic topological entropy relative to $\Sigma$ is \[ h_\Sigma(G\curvearrowright X) =\sup_{\varepsilon>0}\ \inf_{\substack{F\subseteq G\text{ finite}\\F\ne\varnothing}}\ \inf_{\delta>0} \limsup_{i\to\infty}\frac1{d_i}\log N_\varepsilon\left(\operatorname{Map}(\rho,F,\delta,\sigma_i),\rho_\infty\right), \qquad \log0=-\infty. \] Question under these conventions. Do there exist $G$, a sofic approximation $\Sigma$ of $G$, and a distal action $G\curvearrowright X$ as above such that \[ h_\Sigma(G\curvearrowright X)=-\infty? \] The existential quantifier over $\Sigma$ is an explicit possible interpretation, not a source-verified qualification. The supplied record does not determine whether a particular approximation is understood, whether the conclusion is required for every approximation, or whether additional ambient hypotheses apply.

Short English statement

Can a distal action have sofic topological entropy $-\infty$? A precise version in the standard compact metrizable setting is given above, but the intended group, space assumptions, and quantification over sofic approximations require source confirmation. The supplied record labels the question open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002426$, title: Distal Actions with Negative Infinite Sofic Entropy; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. No individual proposer is supplied.
• Supplied source citation: Arbeitsgemeinschaft: Sofic Entropy (2014), Oberwolfach Reports, DOI 10.4171/owr/2013/50, pp. 2919--2961. https://doi.org/10.4171/owr/2013/50. This same link is supplied in literature_sources under the label "original OWR report." The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature assessment dated 2026-08-21: the primary source is reported still to state that it is unknown whether a distal action can have sofic topological entropy $-\infty$, and the curation reports locating no later resolution. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports comparison with the official Oberwolfach report. Those are supplied verification claims; the report has not been independently consulted for this draft. The citation names 2014, whereas the proposed year and year tag are 2013; this discrepancy remains unresolved.