Hyperbolic Three-Manifolds

Of the eight geometries, the hyperbolic one is the richest and the most common. Most knot complements, most Dehn fillings and, in a precise sense, most 3-manifolds are hyperbolic. It is also the geometry the Ricci flow reaches at the end of time. After surgeries, the parts of a 3-manifold that do not become extinct or collapse expand, and once rescaled by $\frac1t$ they converge to hyperbolic metrics (10A.7 Geometrization and Ricci Flow, 12C.4 Geometrization). This chapter shows how hyperbolic structures are built from ideal tetrahedra, why they are unique (Mostow rigidity), and how every hyperbolic manifold splits into a thick part and a thin part made of tubes and cusps.

By the end of this chapter you will be able to:

• describe $\mathbb{H}^3$ in the upper half-space model and its isometries as Möbius transformations;
• compute the volume of an ideal tetrahedron with the Lobachevsky function;
• build the figure-eight knot complement from two regular ideal tetrahedra and compute its volume;
• state Mostow rigidity and explain why volume is a topological invariant;
• state the Margulis lemma, describe the thick–thin decomposition, and state hyperbolic Dehn filling.

6.1Computing hyperbolic structures

6.2Hyperbolic space and its isometries

In the upper half-space model $\mathbb{H}^3 = \{(z, t) : z \in \mathbb{C}, t > 0\}$ with metric $\frac{|dz|^2 + dt^2}{t^2}$ (9A.1 Riemannian Metrics and Model Spaces), the boundary at infinity is the Riemann sphere $\hat{\mathbb{C}} = \mathbb{C}\cup\{\infty\}$. Every Möbius transformation $z \mapsto \frac{az + b}{cz + d}$ of $\hat{\mathbb{C}}$ extends uniquely to an isometry of $\mathbb{H}^3$ (the Poincaré extension), and the orientation-preserving isometry group is

A closed (or finite-volume) orientable hyperbolic 3-manifold is $\mathbb{H}^3/\Gamma$ for a discrete, torsion-free subgroup $\Gamma \subset PSL(2, \mathbb{C})$, a Kleinian group. Geodesics are vertical lines and semicircles orthogonal to $\hat{\mathbb{C}}$; totally geodesic planes are vertical half-planes and hemispheres; horospheres are horizontal planes $t = c$ and spheres tangent to $\hat{\mathbb{C}}$, and the region above a horosphere is a horoball.

6.3Ideal tetrahedra

An ideal tetrahedron is the convex hull of four distinct points of $\hat{\mathbb{C}}$: a tetrahedron with all its vertices at infinity, and still finite volume. The dihedral angles at opposite edges are equal, and the three angles $\alpha, \beta, \gamma$ at a vertex satisfy $\alpha + \beta + \gamma = \pi$, because a horosphere centred at an ideal vertex cuts the tetrahedron in a Euclidean triangle with those angles. Milnor showed that the volume is

the Lobachevsky function (Figure 6.1). Volume is maximised by the regular ideal tetrahedron, with all angles $\frac{\pi}{3}$ (Exercise 6.4):

6.4The figure-eight knot complement

Robert Riley showed in 1975 that the complement of the figure-eight knot admits a complete hyperbolic structure, by finding a discrete faithful representation of its group into $PSL(2, \mathbb{C})$. Thurston then saw it geometrically: the complement is obtained by gluing the faces of two regular ideal tetrahedra in pairs. Around each of the two resulting edges, six dihedral angles of $\frac{\pi}{3}$ meet, totalling $2\pi$, so the gluing is consistent, and the structure is complete (Exercise 6.5). Hence

the number SnapPy prints. In general one solves gluing equations: each ideal tetrahedron has a complex shape parameter, and the product of the parameters around each edge must equal $1$, so that the angles around the edge sum to $2\pi$ and there is no twist.

6.5Mostow rigidity

George Mostow proved this for closed manifolds in 1968 (in all dimensions $n \geq 3$), and Gopal Prasad extended it to finite volume in 1973. The theorem fails in dimension two, where a surface of genus $g \geq 2$ has a $(6g - 6)$-dimensional space of hyperbolic metrics. In dimension three the hyperbolic metric is determined by the topology. So every geometric invariant of a hyperbolic 3-manifold, such as its volume, the lengths of its closed geodesics or the shapes of its cusps, is a topological invariant. This is why the Ricci flow can converge to "the" hyperbolic metric on the thick part: there is only one.

6.6The thick–thin decomposition

The thick part $M_{\geq\varepsilon}$ is compact when $M$ has finite volume. Its geometry is controlled: at every point the injectivity radius is at least $\frac\varepsilon2$, and the curvature is $-1$. In a cusp, the horospheres $t = c$ are flat tori that shrink exponentially as $t \to \infty$ (Figure 6.2); the cusp is a collapsed end in the sense of 9B.3 Collapsing and Noncollapsing, with bounded curvature. A Margulis tube is the hyperbolic version of a thin solid torus. Both thin pieces have a torus at their boundary, and in the Ricci flow the thin part becomes a graph manifold, glued to the hyperbolic thick part along these tori (10A.4 Seifert Spaces and the JSJ Decomposition, 12C.4 Geometrization).

Hyperbolic Dehn filling. Thurston proved that if $M$ is a hyperbolic manifold with a cusp, then all but finitely many Dehn fillings of the cusp (10A.2 Building Three-Manifolds) produce closed hyperbolic manifolds, whose volumes are less than $\operatorname{Vol}(M)$ and converge to it. The filled-in solid torus contains a short geodesic, the core of a thin Margulis tube. So hyperbolic manifolds are abundant: from the figure-eight complement alone come infinitely many closed hyperbolic manifolds. The set of volumes of hyperbolic 3-manifolds is well ordered, of order type $\omega^\omega$ (Jørgensen and Thurston). The smallest closed one is the Weeks manifold, with volume $0.9427\ldots$ (10A.5 Thurston’s Eight Geometries).

6.7History

Robert Riley found the hyperbolic structure on the figure-eight knot complement in 1975. Thurston's Princeton lecture notes (1978–80) introduced ideal triangulations, hyperbolic Dehn filling and the thick–thin picture for 3-manifolds; Milnor's 1982 survey gave the Lobachevsky volume formula its modern form. Mostow's rigidity theorem dates from 1968, Prasad's extension from 1973, and the Margulis lemma from the late 1960s (Kazhdan and Margulis, 1968). Jeffrey Weeks wrote SnapPea in the 1980s and 1990s; Culler and Dunfield's SnapPy built on it from 2009.

6.8Exercises

Solution

$\log|2\sin u|$ is even and $\pi$-periodic, so its integral from $0$ is odd, and periodic once its integral over a period vanishes; $\int_0^\pi\log|2\sin u|\,du = \pi\log2 + \int_0^\pi\log\sin u\,du = \pi\log2 - \pi\log2 = 0$. By symmetry about $\frac{\pi}{2}$, $\Lambda(\frac\pi2) = -\frac12\int_0^\pi\log|2\sin u|\,du = 0$. $2\sin\theta = 1$ at $\theta = \frac{\pi}{6}, \frac{5\pi}{6}$.

Solution

Lagrange: $\Lambda'(\alpha) = \Lambda'(\beta) = \Lambda'(\gamma)$, that is $\log(2\sin\alpha) = \log(2\sin\beta) = \log(2\sin\gamma)$. With all angles in $(0, \pi)$ and summing to $\pi$, equal sines force equal angles (if $\beta = \pi - \alpha$ then $\gamma = 0$). On the boundary (an angle $\to 0$) the volume tends to $0$ (the tetrahedron degenerates), so the interior critical point $\frac\pi3$ is the maximum.

Solution

(a) A small loop around a glued edge passes through the tetrahedra around it; for a smooth hyperbolic structure the total angle must be $2\pi$ (with no rotation along the edge, the second gluing condition). Six angles of $\frac\pi3$ give $2\pi$. (b) The ideal vertices are removed, so the cell structure of the (open) complement has $0$ vertices, $2$ edges, $4$ faces and $2$ cells: $\chi = 0 - 2 + 4 - 2 = 0$. A compact 3-manifold with torus boundary has $\chi = \frac12\chi(\partial M) = 0$, consistent.

Solution

Isometries fixing $\infty$ are $z \mapsto az + b$; preserving the horosphere forces $|a| = 1$, so they act on the plane $t = c$ (with its induced flat metric, scaled by $\frac1c$) by Euclidean isometries. A torsion-free discrete cocompact group of orientation-preserving Euclidean isometries of the plane consists of translations (a nontrivial rotation would have a fixed point; Bieberbach), so it is a lattice $\mathbb{Z}^2$, and the quotient of the horosphere is a flat torus.

Solution

(a) $\operatorname{Ric} = -(n - 1)g$ for curvature $-1$, and $\operatorname{Ric}$ is scale-invariant (9A.4 Curvature and What It Means), so $\partial_t((1 + ct)g_0) = cg_0 = -2\operatorname{Ric}(g_0) = 2(n - 1)g_0$ gives $c = 2(n - 1)$. (b) $\frac{1 + 4t}{t} \to 4$; scaling a metric by $4$ divides curvature by $4$. (c) $\operatorname{Vol}(g(t)) = (1 + 4t)^{3/2}\operatorname{Vol}(g_0)$.