Cusp multipliers detect completeness

The positive shapes determine nondegenerate ideal tetrahedra. The face maps are hyperbolic isometries. Around each finite edge, the edge product $1$ makes the accumulated complex similarity the identity, and the $2\pi$ angle sum removes a cone angle. Thus the tetrahedra glue to a smooth hyperbolic metric away from the ideal vertex. Horospherical truncations give Euclidean triangles on the torus link; their face identifications are Euclidean similarities, maps of the form $x\mapsto aRx+b$ with $a>0$, $R$ a rotation, and have no cone singularities. They define a Euclidean similarity structure: an atlas with these transition maps. Its linear holonomy, the product $aR$ accumulated around a loop, need not yet be trivial.

For a loop $\gamma$ on the link, the linear part of its similarity holonomy is $H(\gamma )$: transporting a tangent vector across each corner multiplies it by the corner parameter or its reciprocal according to the side used. The basis loops have multiplier $1$, so the linear holonomy is trivial on ${\pi }_1(T)$, since every loop is a word in that basis.

Trivial linear holonomy lets a Euclidean metric in one similarity chart be transported consistently to every chart: two transport paths differ by a loop, whose linear holonomy is the identity by step 1.2. The local Euclidean metrics therefore agree on overlaps and give a global flat metric on the compact torus link. A compact Riemannian manifold is complete.

Lift the flat metric of step 2.1 to the simply connected universal cover and integrate its Euclidean charts along paths to obtain a local isometry into ${\mathbb{R}}^2$; simple connectivity makes this construction path-independent up to one rigid motion. Completeness lets the local isometry extend along every straight segment, so it is a covering onto ${\mathbb{R}}^2$ and hence an isometry. Its deck group acts by Euclidean isometries. Every deck transformation has identity linear part by step 1.2, so acts by translation; compactness of the quotient makes this group a rank-two lattice. The cusp is the corresponding horoball quotient.

In horospherical coordinates the quotient end has metric $d{s}^2=d{r}^2+e^{-2r}{g}_T$ for $r\ge 0$, where $g_T$ is the complete flat torus metric from step 2.1. A path escaping toward the ideal vertex has $r\to \mathrm{\infty }$ and length at least the change in $r$, hence infinite length. Thus the glued metric is complete at its only end, the torus cusp.