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Visualization of the isometry group action on the Fomenko-Matveev-Weeks manifold. (English) Zbl 0891.57014

The hyperbolic 3-manifold \({\mathcal M}_1\) of smallest known volume (closed orientable of volume \(0.94\dots)\) was found independently by Fomenko-Matveev and Weeks using computer calculations; it is obtained, for example, by surgery with coefficients \((5,-2)\) and \((5,-1)\) on the two components of the Whitehead link (recently it has been announced that \({\mathcal M}_1\) is the smallest volume arithmetic 3-manifold). Its isometry group is known to be the dihedral group \(\mathbb{D}_6\) of order 12. In the present paper, a detailed description is given of the nine 3-orbifolds which are the quotients of \({\mathcal M}_1\) by the nontrivial subgroups of \(\mathbb{D}_6\) (up to conjugation). For example, \({\mathcal M}_1\) is the 2-fold cyclic branched covering of the 3-sphere along the 3-bridge knot \(9_{49}\), the 3-fold cyclic branched covering of the 2-bridge knot \(5_2\) and the \(2\times 3=6\)-fold cyclic branched covering of the 2-component link \(7^2_1\). It is also a 2-fold branched covering of two lense spaces. The remaining dihedral quotients give the 3-sphere, the singular sets or branch sets are \(\Theta\)-curves resp. of tetrahedral type in the case of the whole group.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57M60 Group actions on manifolds and cell complexes in low dimensions
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