The Bounded-Height Conjecture for Hyperbolic Dehn Fillings

Definitions and mathematical statement

For a torus cusp of a hyperbolic $3$-manifold, a slope is an unoriented isotopy class of essential simple closed curves on a torus cross-section. Dehn filling along that slope attaches a solid torus to the corresponding boundary component of a compact core so that its meridian represents the chosen slope. Filling several torus cusps requires a choice of slope at each cusp being filled.

Bounded-height conjecture, with unresolved conventions. For a $k$-cusped hyperbolic $3$-manifold $M$, let $\mathcal D(M)$ denote the collection of Dehn fillings intended by the source, and let $H_M(N)$ denote the intended numerical height of a filling $N$. These are placeholders for source conventions that are not supplied. The asserted uniform boundedness has the quantifier structure \[ \text{for every such }M,\quad \exists B_M\in{\mathbb{R}}\quad \forall N\in\mathcal D(M),\qquad H_M(N)\le B_M. \] The bound may depend on $M$, but must be independent of the filling slopes. The assertion is to hold for every cusp number $k$ in the intended range.

Unresolved qualifications. The record does not define the height, its underlying algebraic data, or its normalization. In particular, it does not identify whether the height belongs to a holonomy point, an eigenvalue tuple, a trace, or another object; none of these interpretations is selected here. It also does not specify whether all cusps must be filled, whether only hyperbolic fillings are included, or whether sufficiently long slopes are required. Completeness, finite volume, orientability, the restriction to torus cusps, and the permitted range of $k$ are not explicitly stated. The torus-cusp definition above explains the usual filling operation without asserting these missing hypotheses. Consequently, the displayed formula records the intended boundedness claim but is not a fully determined mathematical conjecture until the height and filling conventions are recovered.

Short English statement

The conjecture asks whether each fixed cusped hyperbolic $3$-manifold has a single upper bound for the heights of all its intended Dehn fillings. The supplied record leaves the height and the permitted fillings undefined, so a precise final formulation requires source clarification. Its supplied status is partially solved, a dated curation label rather than an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002646$, title The Bounded-Height Conjecture for Hyperbolic Dehn Fillings; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. No proposer is identified; the supplied proposed year is 2014.
• Supplied source citation: Low-dimensional Topology and Number Theory (2015), Oberwolfach Reports, pp. 2115--2176, DOI 10.4171/owr/2014/38. Original report link: https://doi.org/10.4171/owr/2014/38. The record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Jeon, "bounded height theorem," https://arxiv.org/abs/1409.2069.
• Supplied literature reference: Jeon, "hyperbolic Dehn fillings," https://doi.org/10.1112/plms.70017.
• Literature assessment dated exactly 2026-08-21: the supplied curation reports uniform bounded height for one cusp and important two-cusp cases, including rationally independent cusp shapes, and reports that no general theorem for all $k$-cusped manifolds was found. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. This is supplied registry metadata, not mathematical verification of this draft or independent verification of the references.