Problem 7.14 — (D.

Definitions and mathematical statement

For a finite-volume hyperbolic three-manifold, use hyperbolic volume in curvature $-1$ normalization. For a closed oriented three-manifold $M$, its simplicial volume is \[ \|M\|=\inf\left\{\sum_i|a_i|:\sum_i a_i\sigma_i \text{ is a real singular cycle representing }[M]\right\}; \] other boundary/end settings require the appropriate relative or locally finite convention.

The source asks to construct a root-of-unity-dependent family of three-manifold invariants whose growth recovers hyperbolic volume, or simplicial volume. If $I_r(M;\zeta_r)$ denotes a proposed family at chosen roots, a possible exponential-growth formulation would be \[ \lim_{r\to\infty}\frac{c}{r}\log|I_r(M;\zeta_r)| =\operatorname{Vol}_{\mathrm{hyp}}(M), \] or a normalized version involving $\|M\|$. This is only an illustrative target: the excerpt does not specify exponential versus other growth, normalization $c$, root choice, admissible levels, zero values, or probability-free limit/limsup convention.

The manifold class, invariant construction and any requirement that it be independent of already-known volume are also absent. No particular quantum invariant or modern volume conjecture is substituted. Hyperbolic volume and simplicial volume are retained as the source's two alternatives, with their domains and normalizations to be fixed rather than identified without conditions.

The summary calls this not obviously resolved and gives no precise candidate or theorem. The 2002 request and later open/triage assessment remain qualified source information pending primary-literature review.

Short English statement

Find root-of-unity invariants whose asymptotic growth recovers three-manifold hyperbolic or simplicial volume. The family, level sequence, growth law and normalization are not specified, so no particular volume-limit conjecture is silently selected.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400129$, 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 7.14, attributed to D. Thurston, PDF page 111: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29; provenance distinguishes 231 numbered items.
• No exact candidate family, normalization or current resolution reference is supplied.