VOL3 and other exceptional hyperbolic 3-manifolds (Q2718986)

``D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional hyperbolic manifolds in their proof that a manifold which is homotopy equivalent to a hyperbolic 3-manifold is hyperbolic. These families are each conjectured to consist of a single manifold. In fact, an important point in their argument depends on this conjecture holding for one particular exceptional family. In this paper, we prove the conjecture for that particular family, showing that the manifold known as VOL3 in the literature covers no other manifold. We also indicate techniques likely to prove this conjecture for five of the other six families.'' NEWLINENEWLINENEWLINEThe above important rigidity theorem due to Gabai, Meyerhoff and Thurston has its origin in earlier work of \textit{D. Gabai} [J. Am. Math. Soc. 10, No. 1, 37-74 (1997; Zbl 0870.57014)] where he proved it under the additional hypothesis that some closed geodesic in the hyperbolic 3-manifold satisfies a certain ``insulator condition'' (which holds if there is an embedded hyperbolic tube of radius \((\log 3)/2\) about the geodesic). The proof of the rigidity theorem is achieved by computer showing that, apart from seven exceptional families, all closed hyperbolic 3-manifolds possess such a closed geodesic. The manifold VOL3 has the volume of the regular ideal simplex in hyperbolic 3-space which is the third smallest volume known for a hyperbolic 3-manifold [see e.g. a paper by \textit{C. D. Hodgson} and \textit{J. R. Weeks}, Exp. Math. 3, No. 4, 261-274 (1994; Zbl 0841.57020)]. In the present paper, arithmetic techniques are applied to obtain the uniqueness of the family resp. the properties of the manifold VOL3.