Compact Hyperbolic 3-Manifolds

• Compact hyperbolic 3-manifolds are 3D spaces with complete, finite-volume metrics of constant negative curvature and often include totally geodesic boundary surfaces.
• Their explicit combinatorial decompositions, such as octahedral arrangements and angle structures, provide concrete methods to certify hyperbolic metrics.
• Advanced constructions using handlebodies and polyhedral tessellations yield predictable volume bounds and establish rich connections with algebraic and topological invariants.

A compact hyperbolic 3-manifold is a compact 3-dimensional manifold admitting a complete, finite-volume Riemannian metric of constant sectional curvature $-1$, often with boundary consisting of surfaces of genus at least $2$ that are totally geodesic. The study of such 3-manifolds, particularly those with totally geodesic boundary (here denoted "tg-hyperbolic"), is fundamental in low-dimensional topology, geometric group theory, and the theory of Kleinian groups. Compact hyperbolic 3-manifolds appear in several distinct constructions, admit explicit combinatorial decompositions, and display rich connections to algebraic, topological, and geometric invariants.

1. Definitions and Structural Properties

A compact orientable 3-manifold $M$ is called hyperbolic if, after capping all $S^2$ boundary components with 3-balls and removing open regular neighborhoods of all $T^2$ boundary components, the resulting manifold admits a complete metric of constant negative curvature $-1$ and finite volume. A manifold is tg-hyperbolic if, in such a metric, each boundary component of genus at least $2$ is totally geodesic, that is, locally isometric to $\mathbb{H}^2$ embedded as a hypersurface with vanishing second fundamental form (Adams et al., 31 Jan 2025).

The existence of hyperbolic metrics with totally geodesic boundary is equivalent to the existence of an ideal triangulation supporting an angle structure: an assignment of dihedral angles to corners of truncated hyperideal tetrahedra forming the manifold, satisfying linear edge and face sum constraints (see §3) (Zhang et al., 2014).

2. Explicit Constructions via Handlebody Complements

For any compact orientable 3-manifold $M_0$, after the above normalization, there exist embedded handlebodies $H$ of arbitrary genus $g\geq2$ such that the closure of the complement $\overline{M\setminus H}$ is tg-hyperbolic (Adams et al., 31 Jan 2025). The key construction employs special handle decompositions (following Myers) where 0-handles meet exactly four 1-handles. Spatial graphs formed by inserting "knotted graph tangles" in the 0-handles yield handlebodies whose exteriors, by application of Myers' Gluing Lemma and Thurston's hyperbolization for Haken manifolds, are simple (irreducible, boundary-irreducible, atoroidal), ensuring the existence of a tg-hyperbolic structure.

This procedure leads to the volume spectrum of handlebody exteriors: for each $g\ge1$,

$$v_g^{\min}(M)=\min\left\{\operatorname{Vol}(\overline{M\setminus H})\mid H\subset M,\ \text{genus }g,\ \overline{M\setminus H}\text{ tg-hyperbolic}\right\}$$

which forms a nondecreasing sequence of positive real numbers indexed by $g$, growing linearly with genus.

3. Combinatorial Decomposition: Octahedra and Angle Structures

A central combinatorial tool is the octahedral decomposition of hyperbolic 3-manifold exteriors, originally developed for link complements and adapted to spatial graphs retracting to a handlebody (Adams et al., 31 Jan 2025). Each crossing in a 4-valent diagram of the graph is surrounded by an ideal regular octahedron (or truncated octahedron at trivalent vertices), and the octahedra are assembled face-to-face, matching truncated faces with boundary components. This yields an explicit cellular decomposition of the tg-hyperbolic complement into ideal and truncated octahedra.

Angle structures, as formalized in (Zhang et al., 2014), provide a combinatorial certificate of hyperbolicity with totally geodesic boundary. An assignment $\alpha$ of strictly positive dihedral angles in $(0,\pi)$ to each tetrahedral corner satisfies:

• At every interior edge, the sum of adjacent angles is $2\pi$.
• At every truncation face, incident angles sum to less than $\pi$.

Such an angle structure guarantees the existence (and, via variational principles, uniqueness) of the hyperbolic metric with prescribed combinatorics. Conversely, any hyperbolic 3-manifold with totally geodesic boundary is ideally triangulated by hyperideal and flat tetrahedra supporting an angle structure obtained by deformation of the geometric decomposition.

4. Quantitative Geometry: Volume Bounds and Growth

Hyperbolic volume provides a fundamental invariant of compact 3-manifolds and their submanifolds. For tg-hyperbolic handlebody exteriors of genus $g$, one obtains explicit two-sided linear estimates:

$$(2g-2)\,v_{\mathrm{tet}} \leq \operatorname{Vol}(\overline{M\setminus H_g}) \leq (4g+C)\,v_{\mathrm{oct}}$$

with $v_{\mathrm{tet}} \approx 1.01494$ and $v_{\mathrm{oct}} \approx 3.66386$ the volumes of the regular ideal tetrahedron and octahedron, respectively, and $C$ a universal constant (Adams et al., 31 Jan 2025). The crossing number of the spatial graph grows linearly with genus, and so does the hyperbolic volume, confirming that larger genus handlebodies force at most linear volume growth in their hyperbolic exteriors.

Further, in certain polyhedral and covering constructions (see below), volume and injectivity radius can be controlled explicitly via the geometry of right-angled polytopes, colouring rules, and reflection group covers, leading to infinite families of examples with known minimal volumes and systoles (Kolpakov et al., 2013).

5. Alternative Constructions: Polyhedral Tessellations and Geometric Bounding

Kolpakov, Martelli, and Tschantz utilize right-angled dodecahedra and $120$-cells in higher dimensions to construct explicit infinite families of closed hyperbolic 3-manifolds tessellated by $16k$ right-angled dodecahedra ($M_k$), each bounding a compact hyperbolic 4-manifold tessellated by $32k$ right-angled $120$-cells ($W_k$). The combinatorial technique uses a Davis–Januszkiewicz colouring of facets by elements of $(\mathbb{Z}/2)^s$ to pass to torsion-free covers with prescribed gluing patterns and orientation constraints (Kolpakov et al., 2013).

In these families,

$$\operatorname{Vol}(M_k)=16k\,V_D,\quad \operatorname{Vol}(W_k)=32k\,V_Z$$

where $V_D\approx4.3062$ is the volume of the regular right-angled dodecahedron and $V_Z=\frac{34}{3}\pi^2\approx111.30$ is the regular $120$-cell volume. The ratio $$\operatorname{Vol}(W_k)/\operatorname{Vol}(M_k)=2\,V_Z/V_D\approx51.7$$ is universal, bounding the growth of higher-dimensional fillings.

6. Algebraic and Topological Invariants of Compact Hyperbolic 3-Manifolds

Sectional curvature in a hyperbolic manifold is identically $-1$ everywhere, with totally geodesic boundary components inheriting this curvature and vanishing second fundamental form. In constructions where the boundary is totally geodesic, boundary geodesics hit orthogonally and the normal injectivity radius can be estimated below; this controls metric collar neighborhoods crucial for extending curvature in coning-off procedures (Manning et al., 6 Jan 2026).

Passing to closed pseudomanifolds by coning off boundary components yields spaces carrying locally $CAT(k)$ metrics for some $k<0$, with fundamental groups that are Gromov hyperbolic. In certain examples, locally convex subspaces can be constructed with property (T); notably, the ambient fundamental group is then not cubulable and does not have the Haagerup property (Manning et al., 6 Jan 2026). Splittings over surface subgroups persist in the cone-off, indicating nontrivial decomposability in the sense of group theory.

7. Connections to Higher Dimensions and Open Problems

Some compact hyperbolic 3-manifolds bound geometrically: they are isometric to the totally geodesic boundary of compact hyperbolic 4-manifolds. Infinite families with controlled topological and geometric invariants are constructed using right-angled reflection orbifolds and gluing polytopes along isometric faces (Kolpakov et al., 2013). These constructions raise further questions about the relationship between the geometry of a 3-manifold and its possible 4-dimensional fillings, rigidity phenomena, and the structure of their arithmetic invariants.

The existence of infinite volume spectra, minimal volume bounds, and decomposability into polyhedral blocks point to rich combinatorial and geometric possibilities for compact hyperbolic 3-manifolds and their exteriors. The interplay of triangulation, angle structure, and group action techniques continues to influence advances in the field.