Entropy Rigidity for Hyperbolic Group Actions

Definitions and mathematical statement

Under the usual terminology, a metric space $(X,d)$ is proper if every closed bounded subset is compact, geodesic if every two points are joined by a distance-realizing geodesic segment, and Gromov hyperbolic if its geodesic triangles are $\delta$-thin for some $\delta\geq 0$. A geometric action of a discrete group $\Gamma$ means a proper, cocompact action by isometries. Write $\partial_\infty X$ for the Gromov boundary. The supplied hypothesis is \[ \partial_\infty X\cong S^n \] as topological spaces. Here $n$ is conventionally a nonnegative integer; the source record does not explicitly specify its range.

For a basepoint $o\in X$, use the usual orbit-counting definition of the critical exponent: \[ e(\Gamma;X,d)=\limsup_{R\to\infty}\frac{1}{R}\log\#\{\gamma\in\Gamma:d(o,\gamma o)\leq R\}. \] This quantity is independent of the basepoint and is denoted by $e(\Gamma)$ in the supplied statement. This explicit definition uses the usual discrete-group interpretation of a geometric action, which the record does not separately specify.

Under the additive-error convention, a rough isometry from $X$ to a metric space $Y$ is a map $f:X\to Y$ for which there exist constants $A,B\geq0$ such that \[ \left|d_Y(f(x),f(x'))-d(x,x')\right|\leq A\qquad(x,x'\in X), \] and every point of $Y$ lies within distance $B$ of $f(X)$. Write $\mathbb H^{n+1}$ for real hyperbolic space, conventionally normalized to sectional curvature $-1$.

Question as supplied. For every proper geodesic Gromov hyperbolic space $X$ with a geometric action by $\Gamma$ and $\partial_\infty X\cong S^n$, is it true that \[ e(\Gamma)\geq n, \qquad e(\Gamma)=n\quad\Longleftrightarrow\quad X\text{ is rough-isometric to }\mathbb H^{n+1}? \] Unresolved qualifications. The supplied curation explicitly says that surrounding hypotheses needed to identify the intended entropy equality question are missing. In particular, the record gives no metric-scale normalization or additional curvature hypothesis. The numerical critical exponent depends on metric scale, so the bound and equality target require clarification. The conventions above make the extracted question readable but do not supply those missing hypotheses or establish a corrected formulation.

Short English statement

The extracted question asks whether a geometric group action on a proper geodesic hyperbolic space with boundary $S^n$ has critical exponent at least $n$, with equality precisely when the space is rough-isometric to $\mathbb H^{n+1}$. Missing surrounding hypotheses, especially metric normalization, prevent treating this as a fully specified intended conjecture. The supplied open label is dated curation, not an independently verified status.

Sources

• Local registry: supplied UnsolvedMath record, permanent ID $30001822$, source problem code catalogue entry, title Entropy Rigidity for Hyperbolic Group Actions, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2011.
• Supplied report citation: Differentialgeometrie im Großen (2012), Oberwolfach Reports, pp. 1857--1912, DOI 10.4171/owr/2011/33. Supplied source URL and literature-source link, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2011/33. The record describes these reports as documenting mathematical workshops and their open-problem discussions. No exact page or passage for this question is supplied.
• Literature assessment dated 2026-08-21: supplied dated curation, not independently verified. It labels the problem open and states that the entropy equality question was extracted without surrounding hypotheses needed to identify a theorem, and that the cited report does not resolve it in this generality. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. This supplied editorial label does not resolve the missing mathematical hypotheses. The report citation has not been independently consulted for this draft.