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

Ratcliffe, John G.; Tschantz, Steven T.

Exp. Math. 9, No. 1, 101-125 (2000).

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.

Reviewer: Peter B.Gilkey (Eugene)

MSC:

57N13	Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M50	General geometric structures on low-dimensional manifolds

Keywords:

hyperbolic manifolds; \(4\)-manifolds; volume; \(24\)-cell

References:

[1]	Bianchi L., Math. Ann. 40 pp 332– (1892) · JFM 24.0188.02 · doi:10.1007/BF01443558
[2]	Brunner A. M., Trans. Amer. Math. Soc. 282 (1) pp 205– (1984) · doi:10.1090/S0002-9947-1984-0728710-2
[3]	Coxeter H. S. M., Proc. Roy. Soc. London. Ser. A. 201 pp 417– (1950) · Zbl 0041.47202 · doi:10.1098/rspa.1950.0070
[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 · doi:10.1007/BF01212693
[6]	Gibbons G. W., Nuclear Phys. B 472 (3) pp 683– (1996) · Zbl 0925.83018 · doi:10.1016/0550-3213(96)00207-6
[7]	Gromov M., Publ. Math. Inst. Hautes Études Sci. 56 pp 5– (1982)
[8]	Hantzsche W., Math. Ann. 110 pp 593– (1935) · Zbl 0010.18003 · doi:10.1007/BF01448045
[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 · doi:10.1090/S0273-0979-1982-14958-8
[12]	Newman M., Integral matrices (1972) · Zbl 0254.15009
[13]	Ratcliffe J. G., Foundations of hyperbolic manifolds (1994) · Zbl 0809.51001 · doi:10.1007/978-1-4757-4013-4
[14]	Ratcliffe J. G., Classical Quantum Gravity 15 (9) pp 2613– (1998) · Zbl 0939.53037 · doi:10.1088/0264-9381/15/9/009
[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 · doi:10.1515/9781400865321
[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 · doi:10.1017/S0305004100055250
[20]	Zimmermann B., Monatshefte Math. 110 (3) pp 321– (1990) · Zbl 0717.57005 · doi:10.1007/BF01301685