manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
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 of hyperbolic 3-space by the action of a torsion-free discrete group acting by isometries.
For the moment see at volume conjecture.
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
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 , JHEP04(2015) 091 (arXiv:1409.6206)
Dongmin Gang, Nakwoo Kim, Large 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:
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