Volume Envelopes of Cusped Hyperbolic Three-Manifolds

Definitions and mathematical statement

The supplied record denotes by $\Omega_{\mathrm{cm}}(t)$ an "asymptotic envelope function of normalized volumes of cusped hyperbolic $3$-manifolds." It does not define the normalization, the envelope construction, the parameter $t$, or its domain. It also does not specify the class of manifolds used in constructing the envelope, including completeness, finite-volume and orientability requirements.

The requested problem is to establish an explicit upper bound for this function. To express the form of the requested inequality without inventing its missing definitions, let $D$ stand for the intended parameter domain and let $A\subseteq D$ stand for the intended range of comparison. An upper bound would take the form of an explicitly specified function $U\colon A\to{\mathbb{R}}$ satisfying \[ \Omega_{\mathrm{cm}}(t)\leq U(t)\qquad\text{for every }t\in A. \] Here $D$, $A$, and $U$ are placeholders for a formulation, not objects defined by the supplied source. In particular, the record does not determine whether the inequality is required throughout the parameter domain or only in a specified asymptotic regime, nor what dependence of $U$ is permitted. A precise quantified problem therefore requires the missing definition of $\Omega_{\mathrm{cm}}$ and its associated conventions.

The supplied background attributes a lower bound to Adams, but supplies neither its formula nor its hypotheses; it cannot be used here to reconstruct the envelope definition.

Short English statement

The problem asks for an explicit upper bound on the volume envelope $\Omega_{\mathrm{cm}}(t)$ for cusped hyperbolic $3$-manifolds. The supplied record leaves the normalization, envelope construction, parameter domain, and asymptotic comparison range undefined, so the displayed inequality is a formulation template requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003564$, title "Volume Envelopes of Cusped Hyperbolic Three-Manifolds," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2017 and identifies no proposer.
• Supplied report citation: Low-dimensional Topology and Number Theory (2018), Oberwolfach Reports, pp. 2363--2426, DOI 10.4171/owr/2017/38. Supplied source and literature-source link: https://doi.org/10.4171/owr/2017/38. The literature-source label is "Low dimensional Topology and Number Theory." The report contents have not been independently consulted for this draft.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports that the report gives an Adams lower bound for the cusped volume envelope, and states that no general explicit upper bound resolving the question was located. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.