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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:

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