Rigidity from Coarse Embeddings of Warped-Cone Levels

Definitions and mathematical statement

The supplied question is: "Does a rigidity theorem hold for quasi-isometric embeddings, or even coarse embeddings, between corresponding warped-cone level sequences?" The record does not specify the actions, the correspondence between levels, or the conclusion meant by rigidity. Consequently, it does not determine a single fully quantified conjecture.

The following definitions describe the terminology conditionally; they do not supply missing hypotheses for the intended problem. In a customary warped-cone construction, let a finitely generated group $\Gamma$, with finite symmetric generating set $S$, act by homeomorphisms on a compact metric space $(X,d_X)$. For $t>0$, the warped metric $d_{\Gamma,t}$ is the largest metric on $X$ satisfying \[ d_{\Gamma,t}(x,x')\leq t\,d_X(x,x'),\qquad d_{\Gamma,t}(x,sx)\leq 1 \quad(x,x'\in X,\ s\in S). \] The metric space $(X,d_{\Gamma,t})$ is a level of the warped cone. A second action $\Lambda\curvearrowright(Y,d_Y)$ gives levels $(Y,d_{\Lambda,u})$. This is a conventional model for the terminology; the source does not specify whether this model or additional restrictions on the spaces and actions are intended.

For metric-space sequences $(X_n,d_n)$ and $(Y_n,e_n)$, a uniform family of quasi-isometric embeddings consists of maps $f_n:X_n\to Y_n$ and constants $A\geq1$, $B\geq0$, independent of $n$, such that \[ A^{-1}d_n(x,x')-B \leq e_n(f_n(x),f_n(x')) \leq A d_n(x,x')+B \quad(n\in{\mathbb{N}},\ x,x'\in X_n). \] A uniform family of coarse embeddings consists of maps $f_n:X_n\to Y_n$ and nondecreasing functions $\rho_-,\rho_+:[0,\infty)\to[0,\infty)$, independent of $n$, with $\lim_{r\to\infty}\rho_-(r)=\infty$, such that \[ \rho_-(d_n(x,x')) \leq e_n(f_n(x),f_n(x')) \leq \rho_+(d_n(x,x')) \quad(n\in{\mathbb{N}},\ x,x'\in X_n). \] Neither definition requires coarse surjectivity.

Under these conventions, the question asks what rigidity conclusion, if any, follows from such embeddings when \[ (X_n,d_n)=(X,d_{\Gamma,t_n}),\qquad (Y_n,e_n)=(Y,d_{\Lambda,u_n}),\qquad t_n\to\infty,\quad u_n\to\infty. \] The use of uniform controls and diverging level parameters here explains a possible sequence-level interpretation; neither is explicitly quantified in the supplied question. The relationship between $t_n$ and $u_n$, the admissible actions, and the proposed rigidity conclusion remain unresolved. No equivalence of actions, relation between groups, or other specific conclusion can be asserted from this record alone.

Short English statement

The question asks whether embeddings between sequences of warped-cone levels force a rigidity relation between the underlying geometric or dynamical data. It asks about both quasi-isometric and coarse embeddings. The supplied record leaves the necessary hypotheses, the meaning of corresponding levels, and the intended rigidity conclusion unspecified, so this is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30003783$, exact source problem code catalogue entry, title "Rigidity from Coarse Embeddings of Warped-Cone Levels," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified. The supplied proposed year is $2018$.
• Supplied report citation: Mini-Workshop: Superexpanders and Their Coarse Geometry (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/19, pp. 1117--1160. https://doi.org/10.4171/owr/2018/19. The supplied background attributes the question to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source, "workshop report": https://ems.press/content/serial-article-files/46739.
• Supplied literature source, "measured-equivalence consequences": https://arxiv.org/abs/2004.10000.
• Supplied literature source, "dynamical relation from warped cones": https://doi.org/10.1142/S1793525325500128.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. The supplied status is "partially solved." The assessment reports stable-product and uniform-measured-equivalence consequences, and examples for quasi-isometric embeddings, under extra hypotheses; it also reports that no general quasi-isometric or coarse-embedding rigidity theorem is established. The background describes this as dated literature triage derived from the supplied Oberwolfach Reports curation and requests independent verification of time-sensitive status and cited claims.