Hyperbolic volume

This article deals with the volume of hyperbolic manifolds, see also Volume of Hyperbolic Simplices .

In topology , a branch of mathematics , the hyperbolic volume is the volume of a hyperbolic manifold . (The hyperbolic volume of a knot or a loop is often referred to , which means the hyperbolic volume of the complement .)

Hyperbolic volume is a topological invariant because, according to Mostow-Prasad's rigidity, there can at most be one hyperbolic metric of finite volume on a manifold of dimension . ${\displaystyle \geq 3}$

Any dimensions

Surfaces

On a closed, orientable surface of gender , the hyperbolic metric is not unambiguous, but there is a -dimensional module space of hyperbolic metrics, the so-called Teichmüller space . But it follows from the Gauss-Bonnet theorem that all of these metrics have the same volume ${\displaystyle g\geq 2}$ ${\displaystyle (6g-6)}$

${\displaystyle \operatorname {Area} (S_{g})=4\pi (g-1)}$

to have. In particular, the hyperbolic volume is also a topological invariant in dimension 2, although Mostow's law of rigidity does not apply in this dimension.

Even dimensions

From the Gauss-Bonnet-Chern theorem it follows that the hyperbolic volume of even-dimensional manifolds is proportional to the Euler characteristic with a proportionality factor that only depends on the dimension. The factor is a rational multiple of . For example for hyperbolic 4-manifolds one has the formula . ${\displaystyle \pi ^{{\frac {1}{2}}\operatorname {dim} (M)}}$ ${\displaystyle \operatorname {Vol} (M)={\frac {4}{3}}\pi ^{2}\chi (M)}$

Odd dimensions

The hyperbolic volumes also form a discrete subset of the real numbers for odd dimensions (with the exception of dimension 3).

Topological definitions

From the Mostow-Prasad rigidity theorem it follows that hyperbolic volume is a topological invariant. Gromow gave a first topological definition with the simplicial volume , which is defined for any manifolds and in the case of hyperbolic manifolds (except for a factor that only depends on the dimension) gives the volume.

Other topological definitions use the Bloch group or the homology of the isometric group of hyperbolic space.

The volume conjecture establishes a connection between hyperbolic volume and quantum invariants of nodes, which has so far only been proven in a few cases.

Number theoretic properties

In the 3-dimensional case the volume can also be calculated with the help of the Bloch group and in this way it is obtained in particular that hyperbolic volumes of 3-manifolds can always be represented as sums of Bloch-Wigner dilogarithms of algebraic numbers . Analogous conjectures (with suitable variants of the polylogarithm) also exist in higher odd dimensions, while in even dimensions hyperbolic volumes are always rational multiples of powers of . ${\displaystyle \pi }$

The volume of arithmetic hyperbolic manifolds can be determined using Prasad's volume formula .

3-manifolds

Manifolds of finite volume

From Margulis' lemma it follows that an orientable, complete, hyperbolic 3-manifold of finite volume is the union of a compact submanifold bounded by tori and a finite set of peaks (quotients of horoballs modulo effects). ${\displaystyle \mathbb {Z} \oplus \mathbb {Z} }$

Jørgensen's theorem

The hyperbolic volumes of 3-manifolds form a well-ordered subset of the real numbers, i.e. H. every family of hyperbolic 3-manifolds has a least-volume element. There are only finitely many 3-manifolds with the same volume.

For every constant there are only finitely many homeomorphy types of the thick part for complete hyperbolic 3-manifolds of volume . So there is a tangle , all the complete hyperbolic 3-manifolds that the volume by Dehn surgery to be gained. ${\displaystyle C>0}$ ${\displaystyle M_{(\epsilon ,\infty )}}$ ${\displaystyle M}$ ${\displaystyle \leq C}$ ${\displaystyle L_{C}\subset S^{3}}$ ${\displaystyle \leq C}$ ${\displaystyle L_{C}}$

Stretching surgery

Let it be a non-compact hyperbolic 3-manifold of finite volume, for example the complement of a hyperbolic link . The manifolds obtained by gluing full gates to the edge components are referred to as expansion filling . (In the case of a nodal complement, this corresponds to the result of a stretching surgery .) A Thurston theorem says that almost all stretching fillings of a hyperbolic manifold are hyperbolic again. The following applies to the volumes of the manifolds constructed by expansion-filling ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle M}$

${\displaystyle \operatorname {Vol} (M)<\operatorname {Vol} (N)}$

and the sequence of these volumes converges to . Similar to Jørgensen's theorem, one can prove that for every constant there is a finite set of hyperbolic manifolds, so that all hyperbolic manifolds of volume arise from one of these manifolds by stretch-filling. ${\displaystyle \operatorname {Vol} (N)}$ ${\displaystyle C}$ ${\displaystyle M_{1},\ldots ,M_{k}}$ ${\displaystyle \leq C}$

The set of volumes of hyperbolic 3-manifolds therefore has cardinality . There is a smallest volume (the volume of the Weeks manifold ), then volumes , then the first cluster point (the volume of the figure eight - complement ), which is the smallest volume of a non-compact 3-manifold, later than the smallest volume of a manifold with 2 Tips, and so on. ${\displaystyle \omega ^{\omega }}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{1}<x_{2}<x_{3}<\ldots }$ ${\displaystyle x_{\omega }}$ ${\displaystyle x_{\omega ^{2}}}$

Manifolds of smallest volume

Gabai-Meyerhoff-Milley developed mom technology to create complete lists of hyperbolic manifolds of small volume. A mom-n manifold is created by gluing on each of 1 and 2 handles, so that every 2 handle runs over exactly three 1 handles and every 1 handle meets at least two 2 handles. They proved that every hyperbolic 3-manifold of volume has an embedded mom-2 or mom-3 submanifold and is especially created by stretching surgery on a mom-2 or mom-3 manifold. Furthermore, they proved that there are 3 mom-2 and 18 mom-3 manifolds and classified them. In particular, it follows from their work that the volume 0.9427 ... of the Weeks manifold is the smallest possible volume of a hyperbolic 3-manifold. ${\displaystyle T^{2}\times I}$ ${\displaystyle n}$ ${\displaystyle \leq 2{,}848}$

literature

• William P. Thurston : The Geometry and Topology of Three-Manifolds online (Chapters 5.11, 5.12., 6.6, 7)
• Sylvain Maillot: Variétés hyperboliques de petit volume (d'après D. Gabai, R. Meyerhoff, P. Milley, ...). Séminaire Bourbaki. Volume 2008/2009. Exposés 997-1011. Astérisque No. 332 (2010), Exp. 1011, x, 405-417. ISBN 978-2-85629-291-4 pdf

Web links

• David Gabai , Robert Meyerhoff , Peter Milley: Mom technology and hyperbolic 3-manifolds. In the tradition of Ahlfors-Bers. V, 84-107, Contemp. Math., 510, Amer. Math. Soc., Providence, RI, 2010. pdf
• Tudor Dimofte , Sergei Gukov : Topological Quantum Field Theory and the Volume Conjecture pdf
• Ian Agol : Volumes of hyperbolic link complements

Individual evidence

1. ^ Hsien Chung Wang : Topics on totally discontinuous groups. Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969-1970), pp. 459-487. Pure and Appl. Math., Vol. 8, Dekker, New York, 1972.
2. ↑ Alexander Goncharov : Volumes of hyperbolic manifolds and mixed Tate motives. J. Amer. Math. Soc. 12 (1999), no. 2, 569-618. pdf
3. ↑ Thurston, Theorem 5.11.2
4. ↑ Thurston, Theorem 5.12.1
5. ↑ Michael Gromov : Hyperbolic manifolds (according to Thurston and Jørgensen). Bourbaki Seminar, Vol. 1979/80, pp. 40-53, Lecture Notes in Math., 842, Springer, Berlin-New York, 1981. pdf