Geometry and Actions of $\delta$-Median Spaces

Definitions and mathematical statement

Let $X=(X,d)$ denote a metric space. A usual meaning of uniformly locally finite is \[ \forall R>0,\qquad \sup_{x\in X}\#\{y\in X:d(x,y)\le R\}<\infty. \] The supplied record does not specify its convention. It also does not provide the defining axioms of a $\delta$-median space, a strongly bolic metric, or a hierarchically hyperbolic space. These expressions are retained as technical terms from the source; their precise conventions must be recovered before a complete formal statement can be given. In particular, the range of $\delta$, and whether it may change under the constructions below, are unspecified.

The source poses four questions:

1. Given an action $G\curvearrowright X$ on a uniformly locally finite $\delta$-median space, does it give rise to a proper action of the same group $G$ on an $L^p$-space?

   The wording is "any action." The record does not specify the class of groups, whether the given action is isometric, proper, continuous, or cocompact, or whether the resulting action must be affine and isometric. It does not specify the admissible values of $p$, whether $p$ is prescribed or existentially quantified, the underlying measure space, or the meaning of properness for the intended group topology. No additional action hypothesis or quantifier over $p$ is imposed here.

2. Does every uniformly locally finite $\delta$-median space $(X,d)$ admit a strongly bolic metric?

   The record does not specify whether the requested metric is on the underlying set $X$, or what compatibility it must have with $d$, such as inducing the same topology, being quasi-isometric to $d$, or respecting a group action. Thus the admissible replacement metrics remain unspecified.

3. Is every hierarchically hyperbolic space a $\delta$-median space?
4. Is every $\delta$-median space a hierarchically hyperbolic space?

For questions (3) and (4), the source does not repeat the uniform local finiteness assumption from questions (1) and (2). It is therefore not added here. The record does not specify whether the assertions concern the given metric, an equivalent metric, or a quasi-isometry class, nor whether additional structures or bounds are required. Consequently, the abbreviated request that the two classes "coincide" does not yet determine a precise equivalence.

The accompanying discussion attributes to Caprace's appendix to joint work of Iozzi and Fernos, cited there as [3], a fixed-point obstruction for actions of uniform lattices on finite-dimensional $\operatorname{CAT}(0)$ cubical complexes. It attributes to Fioravanti, cited there as [5], an extension to actions on finite-dimensional median spaces. The ambient groups containing these lattices and the action hypotheses are absent, so these are recorded as supplied contextual claims, not formulated as unrestricted theorems. The discussion also states that the class of cocompact median groups differs from the class of $\operatorname{CAT}(0)$ cubical groups, while distinguishing finite-dimensional median groups from finite-dimensional $\operatorname{CAT}(0)$ cubical groups remains unknown. Definitions and hypotheses needed to interpret these group classes are not supplied. This contextual comparison is retained separately from the four enumerated questions.

Short English statement

The source asks whether actions on uniformly locally finite $\delta$-median spaces yield proper $L^p$-actions, whether these spaces admit strongly bolic metrics, and whether $\delta$-median and hierarchically hyperbolic spaces imply one another. Missing definitions, action hypotheses, metric compatibility conditions, and quantifiers over $p$ prevent a fully precise formulation. The supplied open status and contextual fixed-point claims require independent review.

Sources

• Local registry: supplied UnsolvedMath record, permanent ID $30005180$, source problem code catalogue entry, title "Geometry and Actions of $\delta$-Median Spaces." Repository registry locations are data/unsolvedmath/problems.json and catalog/problems.json; no independent file inspection is claimed. No individual proposer is identified.
• Supplied report citation: Topologie (2023), Oberwolfach Reports, pp. 1927--1986, DOI 10.4171/owr/2022/34, https://doi.org/10.4171/owr/2022/34. The literature-source label is "Oberwolfach Report 34/2022." The record also gives proposed year 2022; the relationship between these year labels is not explained.
• Supplied contextual attributions: Caprace's appendix to joint work of Iozzi and Fernos, source-local reference [3], and Fioravanti, source-local reference [5]. Full bibliographic entries and their precise hypotheses are not supplied.
• Literature assessment dated exactly 2026-08-22: supplied status "open." The supplied assessment says that the four questions are presented as open in the report and that later results address special classes without settling the general questions. Its listed source is https://doi.org/10.4171/owr/2022/34. This is supplied dated curation, not independently verified; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance assertions, not verification performed for this draft.