Volumes of hyperbolic three-manifolds

Neumann, Walter D.; Zagier, Don

doi:10.1016/0040-9383(85)90004-7

Cited by 340 publications

(455 citation statements)

References 4 publications

Supporting

Mentioning

445

Contrasting

Unclassified

Order By: Relevance

“…This has been attributed to Thurston (unpublished) by Gross [10]. It also follows easily from [17] (see also [12]). We give a cohomological proof in [16].…”

Section: Proposition 31 β(M ) = σ([M ]) ∈ B(c)mentioning

confidence: 61%

“…It follows from [7] that any non-compact M has a "genuine" ideal triangulation: one for which f is arbitrarily closely homotopic to a homeomorphism ( [7] gives an ideal polyhedral subdivision and some flat simplices may be needed to subdivide the polyhedra consistently into ideal tetrahedra). The ideal simplices can be deformed to give degree one ideal triangulations (based on the same 3-cycle Y ) on almost all manifolds obtained by Dehn filling cusps of M (see e.g., [17]). …”

Section: Conjecture 15 If the Invariant Trace Fieldmentioning

confidence: 99%

See 1 more Smart Citation

Untitled

Neumann

Yang

1995

Electron. Res. Announc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. For any finite volume hyperbolic 3-manifold M we use ideal triangulation to define an invariant β(M ) in the Bloch group B(C ). It actually lies in the subgroup of B(C ) determined by the invariant trace field of M . The Chern-Simons invariant of M is determined modulo rationals by β(M ). This implies rationality and -assuming the Ramakrishnan conjecture -irrationality results for Chern Simons invariants.

show abstract

“…This has been attributed to Thurston (unpublished) by Gross [10]. It also follows easily from [17] (see also [12]). We give a cohomological proof in [16].…”

Section: Proposition 31 β(M ) = σ([M ]) ∈ B(c)mentioning

confidence: 61%

Section: Conjecture 15 If the Invariant Trace Fieldmentioning

confidence: 99%

Untitled

Neumann

Yang

1995

Electron. Res. Announc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is well-known that if the moduli are all in p þ and define a hyperbolic structure on M, then they satisfy a system C of algebraic equations called compatibility equations, and the structure is complete if and only if the moduli satisfy also the so-called completeness equations M. See [10], [8], [7], [9], [1], [5], [4], [3] for details about ideal triangulations of hyperbolic manifolds.…”

Section: Introductionmentioning

confidence: 99%

“…One of the main applications of the technique of ideal triangulations is the proof of the Thurston hyperbolic Dehn filling Theorem ( [10], [8], [7]). The idea of the proof is to start with a geodesic ideal triangulation of a complete finite-volume hyperbolic 3-manifold M, in which the tetrahedra have structures whose moduli satisfy C and M and then to perturb the moduli along the space of solutions of C. Finally, one looks at the completions of the structures obtained in this way near the complete structure.…”

Section: Introductionmentioning

confidence: 99%

“…The geometric meaning of a triangle with negative modulus is that its orientation is reversed, and this produces overlapping phenomena like the one pictured in Figure 2. The problem of whether a choice of moduli in Cnf0; 1g defines a (even incomplete) hyperbolic structure on M is surprisingly di‰-cult if compared with the classical one in which only positive moduli appear, and in general the answer is still unknown (see [7], [8], [9], [5], [4], [3] for details).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Similarity structures on the torus and the Klein bottle via triangulations

Francaviglia

2006

Advg

View full text Add to dashboard Cite

Abstract. In this paper we study the possibility of defining a similarity structure on the torus and the Klein bottle using the combinatorial data of a triangulation. Given a choice of moduli for the triangles of a triangulation of a surface, the problem is to decide whether such moduli are compatible with a global similarity structure on the surface.We study this problem under two di¤erent viewpoints. From one side we look at the combinatorial data of triangulations, and we develop an algorithmic method, which allows us to reduce the general problem to a simpler one, which is easily solved. From the other side we study the problem more algebraically, looking at the properties of the holonomy, and we give a complete characterization of the choices of moduli defining global similarity structures on the torus (or on the Klein bottle).

show abstract

The volume conjecture and topological strings

Dijkgraaf

Fuji²

2009

Fortschritte der Physik

View full text Add to dashboard Cite

In this paper, we discuss a relation between Jones-Witten theory of knot invariants and topological open string theory on the basis of the volume conjecture. We find a similar Hamiltonian structure for both theories, and interpret the AJ conjecture as the D-module structure for a D-brane partition function. In order to verify our claim, we compute the free energy for the annulus contributions in the topological string using the Chern-Simons matrix model, and find that it coincides with the Reidemeister torsion in the case of the figure-eight knot complement and the SnapPea census manifold m009.The hyperbolic three-manifold M is given bywhere Γ is a torsion-free and discrete subgroup of P SL(2; C) with the action (2.2).The volume of an ideal tetrahedron T αβγ is computed directly by using the metric (2.1) [33].The function Λ(θ) is called the Lobachevsky function. The volume formula can also be rewritten in terms of the Bloch-Wigner function D(z)(2.9)The gluing condition along each edge imposes constraints on the face angles of the ideal tetrahedra.(2.10)Furthermore, we should also consider the gluing condition for the boundary of the knot complement. In order to realize the torus as a boundary, the face angles must satisfy the completeness conditions along the meridian and the longitude.Mostow's rigidity theorem [34] implies that these conditions are solved uniquely with Im z i > 0 for all i. By summing all volumes for the ideal tetrahedra, we are able to determine the hyperbolic volume uniquely for each hyperbolic three-manifold.

show abstract

Volumes of hyperbolic three-manifolds

Cited by 340 publications

References 4 publications

Untitled

Untitled

Similarity structures on the torus and the Klein bottle via triangulations

The volume conjecture and topological strings

Contact Info

Product

Resources

About