×

The volume spectrum of hyperbolic 4-manifolds. (English) Zbl 0963.57012

The authors construct complete open hyperbolic 4-manifolds of smallest volume by gluing together the sides of a regular ideal 24-cell which is contained in hyperbolic space. The authors show that the volume spectrum of hyperbolic 4-manifolds is the set of all positive integral multiples of \(4\pi^2/3\). This determines the volume spectrum of open hyperbolic \(4\) manifolds. The paper deals solely with open manifolds and sheds no light on the volume spectrum of closed hyperbolic \(4\)-manifolds.

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
PDF BibTeX XML Cite
Full Text: DOI Euclid EuDML

References:

[1] Bianchi L., Math. Ann. 40 pp 332– (1892) · JFM 24.0188.02
[2] Brunner A. M., Trans. Amer. Math. Soc. 282 (1) pp 205– (1984)
[3] Coxeter H. S. M., Proc. Roy. Soc. London. Ser. A. 201 pp 417– (1950) · Zbl 0041.47202
[4] Davis M. W., Proc. Amer. Math. Soc. 93 (2) pp 325– (1985)
[5] Pricke R., Math. Ann. 38 pp 50– (1891) · JFM 23.0138.04
[6] Gibbons G. W., Nuclear Phys. B 472 (3) pp 683– (1996) · Zbl 0925.83018
[7] Gromov M., Publ. Math. Inst. Hautes √Čtudes Sci. 56 pp 5– (1982)
[8] Hantzsche W., Math. Ann. 110 pp 593– (1935) · Zbl 0010.18003
[9] Hilden H. M., Topology ’90 (Columbus, OH, 1990) pp 133– (1992)
[10] Hopf H., Nachr. Ges. Wiss. Gottingen Math.-Phys. Kl. pp 131– (1926)
[11] Milnor J., Bull. Amer. Math. Soc. (N.S.) 6 (1) pp 9– (1982) · Zbl 0486.01006
[12] Newman M., Integral matrices (1972) · Zbl 0254.15009
[13] Ratcliffe J. G., Foundations of hyperbolic manifolds (1994) · Zbl 0809.51001
[14] Ratcliffe J. G., Classical Quantum Gravity 15 (9) pp 2613– (1998) · Zbl 0939.53037
[15] Thurston W. P., ”The geometry and topology of 3-manifolds” (1979)
[16] Thurston W. P., Three-dimensional geometry and topology 1 (1997) · Zbl 0873.57001
[17] Vinberg E. B., Mat. Sb. (N.S.) 72 pp 471– (1967)
[18] Wang H. C., Symmetric spaces (St. Louis, MO, 1969–1970) pp 459– (1972)
[19] Wielenberg N., Math. Proc. Cambridge Philos. Soc. 84 (3) pp 427– (1978) · Zbl 0399.57005
[20] Zimmermann B., Monatshefte Math. 110 (3) pp 321– (1990) · Zbl 0717.57005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.