nLab hyperbolic 3-manifold

Idea

A Riemannian manifold which is both a 3-manifold and a hyperbolic manifold is a hyperbolic 3-manifold.

Equivalently this is a Riemannian manifold which is isometric to the quotient space $\mathbb{H}^3/ \Gamma$ of hyperbolic 3-space by the action of a torsion-free discrete group $\Gamma$ acting by isometries.

Properties

Volume conjecture

For the moment see at volume conjecture.

Examples

• hyperbolic 3-space

• hyperbolic solid torus

  • BTZ black hole

  • thermal AdS3

• knot complement of hyperbolic knot

Related concepts

• low dimensional topology

• SnapPy

References

General

• William Thurston, Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds, Annals of Math, 124 (1986), 203–246 (jstor:1971277, arXiv:math/9801019)

• William Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle (arXiv:math/9801045)

• William Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 3 (1982), 357-381 (euclid.bams/1183548782)

See also

• Manifold Atlas, Hyperbolic 3-manifolds

• Wikipedia, Hyperbolic 3-manifold

M5-branes wrapped on hyperbolic 3-manifolds

Discussion of black M5-branes wrapped on hyperbolic 3-manifolds:

• Jerome Gauntlett, Nakwoo Kim, Daniel Waldram, Section 3.1 of: M-Fivebranes Wrapped on Supersymmetric Cycles, Phys. Rev. D63 (2001) 126001 (arXiv:hep-th/0012195)

• Aristomenis Donos, Jerome Gauntlett, Nakwoo Kim, Oscar Varelam, Wrapped M5-branes, consistent truncations and AdS/CMT, JHEP 1012:003, 2010 (arXiv:1009.3805)

• Dongmin Gang, Nakwoo Kim, Sangmin Lee, Holography of Wrapped M5-branes and Chern-Simons theory, Physics Letters B Volume 733, 2 June 2014, Pages 316-319 (arXiv:1401.3595)

• Dongmin Gang, Nakwoo Kim, Sangmin Lee, Holography of 3d-3d correspondence at Large $N$, JHEP04(2015) 091 (arXiv:1409.6206)

• Dongmin Gang, Nakwoo Kim, Large $N$ twisted partition functions in 3d-3d correspondence and Holography, Phys. Rev. D 99, 021901 (2019) (arXiv:1808.02797)

• Dongmin Gang, Nakwoo Kim, Leopoldo A. Pando Zayas, Precision Microstate Counting for the Entropy of Wrapped M5-branes (arXiv:1905.01559)

Suggestion that the statement of the volume conjecture is reall8 AdS-CFT duality combined with the 3d-3d correspondence for M5-branes wrapped on hyperbolic 3-manifolds:

• Gang-Kim-Lee 14b, Section 3.2

• Gang-Kim 18, around (21)