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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

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
57S30 Discontinuous groups of transformations
51M10 Hyperbolic and elliptic geometries (general) and generalizations
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References:
[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
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