Cubulability and Residual Finiteness of Hyperbolic Groups

Definitions and mathematical statement

A metric space $(X,d)$ is median if, for every $x,y,z\in X$, the intersection \[ I(x,y)\cap I(y,z)\cap I(z,x),\qquad I(x,y)=\{w\in X:d(x,w)+d(w,y)=d(x,y)\}, \] consists of exactly one point, denoted $m(x,y,z)$. A subset is convex if it contains $I(x,y)$ whenever it contains $x$ and $y$. A halfspace is a nonempty proper subset whose two complementary sides are convex. Two halfspaces $H,K$ are transverse if all four intersections of their complementary sides are nonempty. The rank of $X$ is the supremum of the sizes of finite families of pairwise transverse halfspaces; finite rank means that this supremum is finite.

A group $G$ is word-hyperbolic if it has a finite generating set for which its Cayley graph has $\delta$-thin geodesic triangles for some $\delta\geq 0$. Interpret the actions below as actions by isometries, subject to the source qualification below. An action is cocompact if $X=G K$ for some compact subset $K\subseteq X$. Two possible meanings of properness require distinction:

• compact-set properness: for every compact $K\subseteq X$, the set $\{g\in G:gK\cap K\ne\varnothing\}$ is finite;
• metric properness: for every $x\in X$ and $R\geq 0$, the set $\{g\in G:d(x,gx)\leq R\}$ is finite.

The supplied record does not specify which meaning is intended, or assert that $X$ is a proper metric space. These conventions must therefore be checked before selecting a definitive formulation.

Use the usual geometric interpretation of cocompactly cubulable: admitting a proper, cocompact action by cubical automorphisms on a CAT(0) cube complex. A group $G$ is residually finite if \[ \forall g\in G\setminus\{1\}\ \exists\text{ a finite group }F\ \exists\phi:G\to F\text{ a homomorphism such that }\phi(g)\ne 1. \] Under the unrestricted-field interpretation, $G$ is linear if there exist a field $k$, an integer $n\geq 1$, and an injective homomorphism $G\hookrightarrow\operatorname{GL}_n(k)$. The record specifies neither a field nor a characteristic.

Questions in the supplied curated formulation. For every word-hyperbolic group $G$ admitting a proper, cocompact isometric action on a finite-rank median space $X$, must each of the following conclusions hold?

1. $G$ is cocompactly cubulable.
2. $G$ is residually finite.
3. $G$ is linear.

Here properness remains subject to the explicit unresolved convention above. No completeness, geodesicity, local compactness, or prescribed rank bound for $X$ is supplied.

Source qualifications. The displayed questions follow the supplied statement and clean statement. The original excerpt contains only the cubulability and residual-finiteness questions, with their hypotheses missing; it does not contain the linearity question. The supplied literature assessment discusses residual finiteness for an action on a median space without explicitly retaining the finite-rank hypothesis. Consequently, that assessment does not establish the intended scope of all three questions. The original excerpt then runs into a separate Problem 12 concerning a Cartan--Hadamard question for median spaces, attributed there to Bowditch; that adjacent material is not part of the group-theoretic statement.

Short English statement

The supplied formulation asks whether every word-hyperbolic group acting properly and cocompactly on a finite-rank median space admits a geometric cubical action, is residually finite, and is linear. The original excerpt omits the hypotheses and linearity question, and the intended properness convention needs review. The supplied open status is a dated curation label, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID 30006623, source code catalogue entry, title: Cubulability and Residual Finiteness of Hyperbolic Groups. The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2026; no proposer is identified.
• Supplied report citation: Median Geometry and Applications (2026), Oberwolfach Reports, pp. 487--540, DOI 10.4171/owr/2026/8. Supplied source link: https://doi.org/10.4171/owr/2026/8.
• Supplied literature source, labeled "OWR report": https://ems.press/content/serial-article-files/53603. The supplied assessment identifies Problem 11(2) as asking about residual finiteness, reports that the report presents it as open, and says that no resolution was found in that review.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified here. The background explicitly calls it dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the status open and the statement status corrected_verified. Its verification note says that only formula delimiters, notation, grammar, and typography were repaired and that mathematical content and scope were preserved. These are supplied registry assertions; the discrepancies between the original excerpt and curated statement still require review.