Filling Links and Essential Systole

TL;DR

This work resolves the question of Freedman–Krushkal by proving that every closed orientable $3$-manifold $M$ contains a filling link $L$ whose complement is hyperbolic. The authors construct such links with arbitrarily large essential systole $esssys(L)$ by combining Cooper–Thurston triangulations with a hyperbolic tetrahedron tangle, and they show that large $esssys(L)$ forces injectivity of the induced map on fundamental groups for any $L$-relative $1$-spine. A central tool is the Kapovich–Weidmann theorem on actions on $\mathbb{H}^3$, which implies that for a generating set of size $n$ the image either is free or contains a short element; a sufficiently large $esssys(L)$ excludes the short-element case, yielding filling. The results extend to full rank--$n$ filling links and provide a path to higher-dimensional analogues, establishing a robust connection between hyperbolic geometry, spines, and core topological properties of $3$-manifolds.

Abstract

We answer a question of Freedman and Krushkal, producing filling links in any closed, orientable 3-manifold. The links we construct are hyperbolic, and have large essential systole, contrasting earlier geometric constraints on hyperbolic links in 3-manifolds due to Adams-Reid and Lakeland-Leininger.

Figures (8)

• Figure 1: The five allowable triangulations of links of vertices are obtained by doubling each of these triangulations of a disk.
• Figure 2: An illustration of the Cooper-Thurston triangulation on a single cube. The cube is triangulated by $24$ tetrahedra, two of which are shown in orange.
• Figure 3: Left: The tangle $L_0 \subset T_0$. Right: The intersection of $L_0$ with the face.
• Figure 4: The fundamental domain in $T_0$ for the action of ${\rm {\frak Tet}}$. The vertices are barycenters of a vertex, edge, face, and of $T_0$, and are labeled here by $0$, $1$, $2$, and $3$, respectively.
• Figure 5: Left: Dihedral angles for hyperbolic structure on $\Delta \space \smallsetminus \space L_0$. Right: Partially ideal polyhedron $P$ with ideal vertices obtained by collapsing arcs to a point (illustrated by a dot).

Theorems & Definitions (33)

• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• proof : Proof of Theorem \ref{['thm:main']} assuming Theorems \ref{['thm:big essential systole filling']} and \ref{['thm:unbounded essential systole']}.
• Remark 1.6
• Definition 1: Nielsen Equivalence
• Lemma 2.1
• proof
• Theorem 2.2: kapovichweidmann