Geodesic
A Lower Bound for the Volume of Hyperbolic 3-Manifolds
Canadian journal of mathematics, Tome 39 (1987) no. 5, pp. 1038-1056

Voir la notice de l'article provenant de la source Cambridge University Press

The motivation for this paper was the work of Thurston and Jørgensen on volumes of hyperbolic 3-manifolds. They prove, among other things, that the set of all volumes of complete hyperbolic 3-manifolds is well-ordered. In particular, there is a hyperbolic 3-manifold which has minimum volume among all complete hyperbolic 3-manifolds. Further, there is a minimum volume member in the collection of complete hyperbolic 3-manifolds with one cusp; and similarly for n cusps. Computer studies to date show that the manifold obtained by performing (5,1) Dehn surgery on the figure-eight knot in the 3-sphere is the leading candidate for the minimum volume hyperbolic 3-manifold. Its volume is about 0.98. The leading one-cusp minimum volume candidate is the figure-eight knot complement in the 3-sphere. Its volume is about 2.03. 
Meyerhoff, Robert. A Lower Bound for the Volume of Hyperbolic 3-Manifolds. Canadian journal of mathematics, Tome 39 (1987) no. 5, pp. 1038-1056. doi: 10.4153/CJM-1987-053-6
@article{10_4153_CJM_1987_053_6,
     author = {Meyerhoff, Robert},
     title = {A {Lower} {Bound} for the {Volume} of {Hyperbolic} {3-Manifolds}},
     journal = {Canadian journal of mathematics},
     pages = {1038--1056},
     year = {1987},
     volume = {39},
     number = {5},
     doi = {10.4153/CJM-1987-053-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1987-053-6/}
}
TY  - JOUR
AU  - Meyerhoff, Robert
TI  - A Lower Bound for the Volume of Hyperbolic 3-Manifolds
JO  - Canadian journal of mathematics
PY  - 1987
SP  - 1038
EP  - 1056
VL  - 39
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1987-053-6/
DO  - 10.4153/CJM-1987-053-6
ID  - 10_4153_CJM_1987_053_6
ER  - 
%0 Journal Article
%A Meyerhoff, Robert
%T A Lower Bound for the Volume of Hyperbolic 3-Manifolds
%J Canadian journal of mathematics
%D 1987
%P 1038-1056
%V 39
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1987-053-6/
%R 10.4153/CJM-1987-053-6
%F 10_4153_CJM_1987_053_6

[1] 1. Buser, P., On Cheeger's inequality λ ≧ h2/4, Proc. of Symposia in Pure Math. 36 (1980), 29–77. Google Scholar

[2] 2. Brooks, R. and Matelski, J., Collars in Kleinian groups, Duke Math. J. 43 (1982), 163–182. Google Scholar

[3] 3. Gallo, D., A, 3-dimensional hyperbolic collar lemma, in Kleinian groups and related topics, Lecture Notes in Mathematics 977 (Springer-Verlag, New York, 1983). Google Scholar | DOI

[4] 4. Halpern, N., A proof of the collar lemma, Bull. Lond. Math. Soc. 13 (1981), 141–144. Google Scholar

[5] 5. Jorgensen, T., On discrete groups of Moebius transformations, Amer. J. Math. 98 (1976), 739–749. Google Scholar

[6] 6. Keen, L., Collars on Riemann surfaces, Ann. of Math. Studies 79 (1974), 239–246. Google Scholar

[7] 7. Kazhdan, D. and Margulis, G., A proof of Selberg's hypothesis, (Russian) Mat. Sb. (N.S.) 75(117) (1968), 163–168. Google Scholar

[8] 8. Lehner, J., A short course in automorphic functions, (Holt, Reinhart and Winston, New York, 1966). Google Scholar

[9] 9. Meyerhoff, R., A lower bound for the volume of hyperbolic 3-orbifolds, preprint. Google Scholar

[10] 10. Matelski, J., A compactness theorem for Fuchsian groups of the second kind, Duke Math J. 43 (1976), 829–840. Google Scholar

[11] 11. Randol, B., Cylinders in Riemann surfaces, Comm. Math. Helvetici 54 (1979), 1–5. Google Scholar

[12] 12. Thurston, W., The geometry and topology of 3-manifolds, Princeton University preprint (1978). Google Scholar

[13] 13. Waterman, P., An inscribed ball for Kleinian groups, Bull. London Math. Soc. 16 (1984), 525–530. Google Scholar

Cité par Sources :