Exceptional regions and associated exceptional hyperbolic 3-manifolds

Details Exceptional regions and associated exceptional hyperbolic 3-manifolds Exceptional regions and associated exceptional hyperbolic 3-manifolds

2003-11-21

arxiv.org/abs/math/0311389v2

Experiment. Math. 16 (2007), no. 1, 107- 118

Mathematics

Geometric Topology

19 pages, minor revisions, added appendix by Alan Reid, published by Experimental Mathematics

Scientific paper

A closed hyperbolic 3-manifold is exceptional if its shortest geodesic does not have an embedded tube of radius $\ln(3)/2$. D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional manifolds in their proof of the homotopy rigidity theorem. They identified the hyperbolic manifold known as Vol3 in the literature as the exceptional manifold associated to one of the families. It is conjectured that there are exactly 6 exceptional manifolds. We find hyperbolic 3-manifolds, some from the SnapPea's census of closed hyperbolic 3-manifolds, associated to 5 other families. Along with the hyperbolic 3-manifold found by Lipyanskiy associated to the seventh family we show that any exceptional manifold is covered by one of these manifolds. We also find their group coefficient fields and invariant trace fields.

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