Problem 12.19 — (J.

Definitions and mathematical statement

In the classical surface setting, a measured geodesic lamination is a closed union of disjoint complete geodesics equipped with locally finite transverse measures invariant under passage along leaves. Write $\mathcal{ML}(S)$ for the resulting measured-lamination space in a specified surface category.

The source asks to extend measured-lamination theory to higher-rank groups. It does not state which real/complex or algebraic Lie groups, which surfaces, or which classical structures the extension must preserve. A proposed family of higher-rank objects would ordinarily need a precise construction and comparison with the rank-one theory, but coordinates, intersection functions, representation-space compactifications, positivity or mapping-class actions are different possible aims. None is imposed as the unique definition of the requested extension.

The source supplies no universal characterization, required theorem or equivalence notion, so this remains a research program rather than one fully quantified conjecture. The summary mentions Fock--Goncharov-related work without exact constructions or hypotheses. Such a mention is not treated as resolving every meaning of higher rank. The 2002 question and later open/triage label remain source information pending current primary-literature review.

Short English statement

Develop a higher-rank-group analogue of measured laminations. The groups, surface class and structures to generalize are not specified, and the cited related work does not select a unique completed theory.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400224$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 12.19, attributed to J. Roberts, PDF page 167: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29.
• Fock and Goncharov are mentioned by the summary without full references or an exact higher-rank comparison theorem.