Unspecified Higher-Rank Dynamics Conjecture

Definitions and mathematical statement

Unrecoverable mathematical statement. The supplied local registry record for permanent UnsolvedMath ID $30004933$, source code catalogue entry, is titled "Unspecified Higher-Rank Dynamics Conjecture." Its extract begins with the isolated word "conjecture" and continues with workshop background. It supplies no complete conjectural assertion, specified mathematical objects, hypotheses, conclusion, or quantifiers from which a precise problem can be recovered. Accordingly, no mathematical conjecture is asserted in this draft, and no definitions can be selected as the intended problem's definitions without additional evidence.

Supplied verification qualification. The record has statement_status: unrecoverable. Its supplied verification note reports that the official report identifies this passage as introductory context pointing to the higher-rank rigidity conjecture retained in OWR-8415361-002; no separate clean statement is promoted, and the original record remains publicly visible for provenance. This is a supplied verification claim, not an independent verification here. The statement of that other record is not supplied and is not substituted for this record.

Context retained without promotion to a conjecture. The passage reports results of Thi Dang on topological mixing of positive diagonal flows and results of Jialun Li on counting and equidistribution of torus orbits for Weyl chamber flows on finite-volume locally symmetric spaces of rank $k>2$. This rank restriction belongs to that contextual discussion; it does not supply hypotheses for the missing conjecture.

The passage describes Anosov representations as providing a class of noncompact locally symmetric spaces of higher rank and as a generalization of convex cocompact manifolds of higher rank. It attributes pioneering work to Labourie and Guichard--Wienhard and mentions geometric, Lie theoretic, dynamical, and projective geometric characterizations of holonomy representations. These are source descriptions whose precise scope and hypotheses are not supplied. Fanny Kassel's minicourse is described as an introduction to these viewpoints. Anna Wienhard is reported to have addressed geometric questions, and Lasse Wolf to have discussed using geometric properties to prove absence of embedded eigenvalues on such spaces. No precise spectral assertion or associated hypotheses are given.

The passage further reports François Ledrappier's discussion of Hausdorff dimensions of limit sets of Anosov representations and their relations with Falconer dimensions. It describes partial encoding of Anosov representations by reparametrizations of the geodesic flow of an abstract Gromov hyperbolic group, underlying uniformly hyperbolic flows, and the use of thermodynamical formalism to define pressure metrics on moduli spaces. It attributes discussion of pressure metrics associated with different linear functionals and their possible degeneracy to Andres Sambarino. It attributes to Xian Dai discussion of a different pressure metric on the Hitchin component in $\mathrm{SL}(3,{\mathbb{R}})$, accessible using microlocal analysis. These topics do not identify the missing conjecture or supply definitions sufficient to formulate one.

Status qualification. The registry labels this item "open." Its supplied literature assessment explains that an unclear, unverified claim was normalized to that label. This is a curation label and does not establish the existence or mathematical open status of a recoverable conjecture.

Short English statement

Permanent UnsolvedMath ID $30004933$, source code catalogue entry, contains workshop background rather than a recoverable conjecture. The supplied verification note points to a higher-rank rigidity conjecture in OWR-8415361-002, whose statement is not provided. The "open" label is supplied curation; a precise mathematical statement requires recovery and human review of the intended source passage.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004933$, source code catalogue entry, title "Unspecified Higher-Rank Dynamics Conjecture," in the Oberwolfach Reports Open Problems collection. The supplied record preserves the original passage for provenance, has no clean statement, and records no proposer. No independent inspection of the local registry is claimed.
• Supplied report citation: Mini-Workshop: $(\mathrm{Anosov})^3$ (2022), Oberwolfach Reports, pp. 3067--3099, DOI 10.4171/owr/2021/56. Source URL: https://doi.org/10.4171/owr/2021/56. The supplied citation says 2022, whereas the proposed year, year tag, and DOI contain 2021; this discrepancy is unresolved.
• Supplied literature source, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2021/56. The literature assessment was checked on 2026-08-22, according to the supplied record. This is supplied dated curation, not independently verified. It labels the item open because the extracted claim is unclear and an unverified claim was normalized as open; the background expressly calls for independent verification of time-sensitive status and cited claims.
• Supplied statement-verification note: the official report is said to identify the extract as introductory context pointing to the higher-rank rigidity conjecture retained in OWR-8415361-002. No separate clean statement is promoted. This attribution is preserved without claiming that the report or the related registry record was visited.
• The passage attributes contextual work and talks to Thi Dang, Jialun Li, Labourie, Guichard--Wienhard, Fanny Kassel, Anna Wienhard, Lasse Wolf, François Ledrappier, Andres Sambarino, and Xian Dai. These mentions do not identify a proposer of this item. An authoritative report reference is supplied, but no complete authoritative statement of a distinct conjecture for this permanent ID is supplied.