5.11 (Futer) — A combinatorial model with explicit bilipschitz constants

Definitions and mathematical statement

A map $\phi:(X,d_X)\to(M,d_M)$ is $L$-bilipschitz if it is a bijection, $1\leq L<\infty$, and \[ L^{-1}d_X(x,y)\leq d_M(\phi(x),\phi(y))\leq Ld_X(x,y) \quad\text{for all }x,y\in X. \] Here $d_M$ is the hyperbolic metric on a $3$-manifold $M$, and $X$ is a metric realization of the proposed combinatorial model.

Construction problem (Futer). Build a combinatorial model for hyperbolic $3$-manifolds together with explicit bilipschitz constants. A precise construction should state its combinatorial data and metric realization $X(M)$, give a bilipschitz comparison $\phi_M:X(M)\to M$, and supply an explicit finite bound $L(M)$ in terms of specified available data. The source does not identify the input, permitted geometric data, model type, or dependence of the constant. In particular, "explicit" is not the same as "universal": the statement does not demand one constant independent of $M$, whereas the supplied research summary describes a search for fully explicit universal constants. That strengthening is not silently imposed.

The excerpt also does not fix completeness, volume, compactness, cusps, boundary, or whether finite combinatorial data suffice. Related triangulation, polyhedral, drilling, cone-manifold, and bounded-geometry constructions mentioned in the summary are context; their additional hypotheses are not adopted as restrictions on the original request.

Short English statement

Construct a combinatorial metric model of hyperbolic $3$-manifolds with a proved, explicitly specified bilipschitz bound. The source does not require the constant to be universal or specify the class of model and allowable input data.

Sources

• Local UnsolvedMath registry, permanent ID 10900036, source code catalogue entry; attribution Futer, subsection 5.11, "A combinatorial model with explicit bilipschitz constants."
• Delp, Hoffoss and Manning, Problems In Groups, Geometry, and Three-Manifolds (2015), subsection 5.11, PDF page 4. Supplied URLs: https://arxiv.org/pdf/1512.04620#page=4 and https://arxiv.org/abs/1512.04620; recorded access 2026-07-29; extraction source-tex checked against rendered PDF text. No source-side resolution is recorded.
• The supplied summary mentions SnapPy, Weeks canonical decomposition, ideal polyhedra and triangulations, quantitative cone-manifold and surgery models, and bounded-geometry drilling/polyhedral models. It provides no full references or theorem supplying the requested construction.
• Supplied status partially_solved and PARTIAL-PROGRESS. The summary says no verified fully explicit universal-constant construction for arbitrary hyperbolic $3$-manifolds was found. That is an editorial search assessment with a stronger uniformity requirement than the short source statement. The 2015 source presents the item as open; no complete modern primary-literature audit is recorded. Status evidence and redistribution terms for the publicly accessible author-submitted TeX and PDF remain NEEDS_REVIEW.