zbMATH — the first resource for mathematics

The smallest arithmetic hyperbolic three-orbifold. (English) Zbl 0643.57011
From the introduction: “In this paper we determine the complete, orientable, arithmetic hyperbolic 3-orbifold \(M_ 0\) of minimal volume. We show, in fact, that \(M_ 0\) has smaller volume than any arithmetic orbifold constructed as an irreducible factor-preserving quotient of the product of some number of upper half planes and half spaces. Our proof is entirely number theoretic, and relies on a formula of A. Borel [Ann. Sc. Norm. Super. Pisa, IV. Ser. 8, 1-33 (1981; Zbl 0473.57003)] for the volumes of such orbifolds. In a later paper, we will apply the same techniques to produce a list of the first few smallest complete orientable arithmetic hyperbolic 3-manifolds.” The orbifold in question is \(M_ 0={\mathbb{H}}\) \(3/\Gamma_ 0\) of volume.039050..., where \(\Gamma_ 0\) is given two descriptions: as the subgroup of PSL(2,\({\mathbb{C}})\) resulting from the group of units in a maximal order in the Hamiltonian quaternion algebra over the field \({\mathbb{Q}}(\sqrt{3+2\sqrt{5}})\); and as the orientation preserving subgroup of the Coxeter group \(\circ -\circ \equiv \circ -\circ\). The equivalence of these descriptions was attributed by Borel [op. cit.], to Thurston, however, A. Reid has pointed out that the second description actually yields a subgroup of index 2 in the desired group. It is striking that \(M_ 0\) remains the likeliest candidate for the smallest orientable hyperbolic 3-orbifold, arithmetic or not. The best known lower bound for this volume is however much smaller: 0.0000017, due to R. Meyerhoff [Comment. Math. Helv. 61, 271-278 (1986; Zbl 0611.57010)], who also gives 0.00082 as a lower bound for the smallest volume of an orientable hyperbolic 3-manifold. The smallest known such manifold is (5,1;5,2)-Dehn surgery on the Whitehead link, of volume.942707..., found by Jeff Weeks. This example is also a candidate for the smallest arithmetic example, which remains elusive, despite the authors’ optimism quoted above.
Reviewer: W.D.Neumann

57N10 Topology of general \(3\)-manifolds (MSC2010)
57S30 Discontinuous groups of transformations
51M10 Hyperbolic and elliptic geometries (general) and generalizations
Full Text: DOI EuDML
[1] Borel, A.: Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Sc. Norm. Super. Pisa, IV. Ser.VIII, 1-33 (1981) · Zbl 0473.57003
[2] Chinburg, T.: A small arithmetic hyperbolic 3-manifold. (To appear in the Proc. Am. Math. Soc.) · Zbl 0621.57006
[3] Chinburg, T.: Volumes of hyperbolic manifolds. J. Differ. Geom.18, 783-789 (1983) · Zbl 0578.22014
[4] Delone, B.N., Faddeev, D.K.: The theory of irrationalities of the third degree. A.M.S. Translations of Mathematical Monographs. Am. Math. Soc.,10 (1964) · Zbl 0133.30202
[5] Godwin, H.J.: On quartic fields with signature one and small discriminants. Q. J. Math. Oxford8, 214-222 (1957) · Zbl 0079.05704
[6] Lang, S.: Algebraic number theory. Reading, Mass.: Addison-Wesley, 1970 · Zbl 0211.38404
[7] Margoulis, G.A.: Discrete groups of isometries of manifolds of nonpositive curvature. Proc. Int. Congr. Math. 1974, Vancouver,2, pp. 21-34
[8] Martinet, J.: Petits discriminants des corps de nombres. In: Armitage, J.V. (ed.), Proceedings of the Journées Arithmétiques 1980. London Math. Soc. Lectures Notes series56, Cambridge University 1982, pp. 151-193 · Zbl 0491.12005
[9] Meyerhoff, R.: A lower bound for the volumes of hyperbolic 3-manifolds. (Preprint 1983) · Zbl 0694.57005
[10] Meyerhoff, R.: The cusped hyperbolic 3-orbifold of minimum volume, research announcement. Bull. Am. Math. Soc.13, 154-156 (1985) · Zbl 0602.57009
[11] Odlyzko, A.M.: Some analytic estimates of class numbers and discriminants. Invent. Math.29, 275-286 (1975) · Zbl 0306.12005
[12] Odlyzko, A.M.: Lower bounds for discriminants of number fields II. Tohoku Math. J.29, 209-216 (1977) · Zbl 0362.12005
[13] Poitou, G.: Sur les petits discriminants. Sém. Delange-Pisot-Poitou (Théorie des nombres), exposé n. 6 (1976/77), pp. 6.01-6.18
[14] Thurston, W.: The geometry and topology of 3-manifolds. Princeton University (preprint 1978) · Zbl 0399.73039
[15] Tits, J.: Travaux de Margulis sur les sous-groups discrets de groupes de Lie. Sém. Bourbaki. Springer Lect. Notes Math.567, 174-190 (1975/76)
[16] Vignéras, M.F.: Arithmétique des algèbres de Quaternions. Springer Lect. Notes Math.800 (1980) · Zbl 0422.12008
[17] Wang, H.C.: Topic on totally discontinuous groups. In: Boothby, W. (ed.), Symmetric spaces, M. Dekker (1972), pp. 460-487
[18] Weeks, J.: Hyperbolic structures on three-manifolds. Princeton Ph. D. thesis (1985) · Zbl 0571.57001
[19] Zimmert, R.: Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung. Invent. Math.62, 367-380 (1981) · Zbl 0456.12003
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.