Dynamical Consequences of Weak Warped-Cone Equivalence

Definitions and mathematical statement

Let $\alpha$ and $\beta$ denote the underlying group actions, and write \[ \mathcal L(\alpha)=\left(L_n(\alpha),d_n^\alpha\right)_{n\in{\mathbb{N}}},\qquad \mathcal L(\beta)=\left(L_n(\beta),d_n^\beta\right)_{n\in{\mathbb{N}}} \] for their respective sequences of warped-cone levels. This notation leaves unspecified the acting groups, the spaces acted upon, the regularity of the actions, the construction of the warped metrics, and the choice of level parameters, since the supplied record does not state these data or hypotheses.

For metric spaces $(U,d_U)$ and $(V,d_V)$, a coarse embedding is a map $f:U\to V$ for which there are nondecreasing functions $\rho_-,\rho_+:[0,\infty)\to[0,\infty)$, with $\lim_{r\to\infty}\rho_-(r)=\infty$, such that \[ \rho_-\left(d_U(u,u')\right) \le d_V\left(f(u),f(u')\right) \le \rho_+\left(d_U(u,u')\right) \quad\text{for all }u,u'\in U. \] For sequences of spaces, one must additionally specify how levels are paired and whether the same control functions work for every paired level.

The source calls two sequences weak-equivalent when subsequences coarsely embed into one another in both directions. Denote this source-described relation by $W$. The record does not determine its full quantifiers: it does not specify the subsequence matching, whether the two directions use the same subsequences, whether controls are uniform across levels, or whether the embeddings concern families of levels or metric spaces formed by their disjoint unions. Thus $W$ is an unresolved placeholder for the intended relation, rather than a completed definition.

Problem. Determine what dynamical information about $\alpha$ and $\beta$ is forced by \[ \mathcal L(\alpha)\;W\;\mathcal L(\beta). \] Schematically, after an admissible class $\mathscr C$ of actions and the precise relation $W$ have been specified, determine which dynamical properties $P$ satisfy \[ \forall\alpha,\beta\in\mathscr C,\qquad \mathcal L(\alpha)\;W\;\mathcal L(\beta) \ \Longrightarrow\ \left(P(\alpha)\Longleftrightarrow P(\beta)\right). \] The source does not specify $\mathscr C$, particular properties $P$, or an equivalence relation on actions that should be recovered. Consequently this is a schematic formulation of the supplied research question, not a uniquely specified conjecture or an assertion of any dynamical consequence.

Short English statement

The question asks which features of group-action dynamics are determined when subsequences of the associated warped-cone levels coarsely embed into one another in both directions. The admissible actions, the exact meaning of embedding sequences, and the intended dynamical conclusions remain unspecified in the supplied record.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30003784$, title Dynamical Consequences of Weak Warped-Cone Equivalence, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified.
• Supplied report citation: Mini-Workshop: Superexpanders and Their Coarse Geometry (2019), Oberwolfach Reports, pp. 1117--1160, DOI 10.4171/owr/2018/19. https://doi.org/10.4171/owr/2018/19. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labelled "workshop report": https://ems.press/content/serial-article-files/46739.
• Supplied literature source, labelled "dynamical relation from warped cones": https://doi.org/10.1142/S1793525325500128.
• Supplied literature source, labelled "space of actions": https://doi.org/10.4171/GGD/942.
• Supplied literature assessment dated 2026-08-21: the record labels the problem partially solved and reports substantial partial results in weak-equivalence theory and warped-cone levelwise quasi-isometry invariants, while describing general recovery of action dynamics from weak-equivalence as unresolved. This is supplied dated curation derived from the OWR collection, not an independently verified assessment. The source itself requests independent verification of time-sensitive status and cited claims.