Exceptional hyperbolic 3-manifolds (Q894381)

An exceptional hyperbolic 3-manifold is a closed hyperbolic 3-manifold which does not have an embedded hyperbolic tube of radius log(3)/2 about its shortest geodesic. In the paper [\textit{D. Gabai} et al., Ann. Math. (2) 157, No. 2, 335--431 (2003; Zbl 1052.57019)], the following ``exceptional manifolds conjecture'' was made : Each exceptional box \(X_{i}, 0\leq i \leq 6\), contains a unique element \(s_{i}\) of a subset \(S=exp(T)\) of a compact region of \({\mathbb C}^{3}\). Further, if \(\{G_{i}, f_{i}, w_{i}\}\) is the marked group associated to \(s_{i}\), then \(N_{i}= {\mathbf H}^{3}/G_{i}\) is a closed hyperbolic 3-manifold with the following properties: (i) \(N_{i}\) has fundamental group \(\langle f, w; r_{1}(X_{i}),r_{2}(X_{i})\rangle\), where \(r_{1}(X_{i}),r_{2}(X_{i})\) are the quasi-relators associated to the box \(X_{i}\). (ii) \(N_{i}\) has a Heegaard genus-2 splitting realizing the above group presentation. (iii) \( N_{i}\) nontrivially covers no manifolds. (iv) \(N_{6}\) is isometric to \(N_{5}\). (v) If \((L_{i},D_{i},R_{i})\) is the parameter in \(T\) corresponding to \(s_{i}\), then \(L_{i},D_{i},R_{i}\) are related as follows: For \(X_{0},X_{5}X_{6}: L=D,R=0\). For \(X_{1},X_{2},X_{3},X_{4}: R=L/2\). Related results to the conjecture have been proved. In the paper under review the authors obtain the following main result: If \(N_{i}\) is an exceptional manifold and \(p: N_{i} \rightarrow M\) is a nontrivial covering projection, then up to conjugacy either \(p: N_{2} \rightarrow m010(-2,3)\) or \(p: N_{4} \rightarrow m371(1,3)\), where the manifolds \(m010(-2,3), m371(1,3)\) are given by SnapPea notation. Furthermore, any two such coverings (for a given domain and range) are topologically conjugate. Finally for each \(p\), \(deg(p)=2\). The authors modify (iii) of the conjecture to the above statement and complete the proof of the conjecture. In the course of proving the result above, they show that if \(N_{i}\not= N_{0}\), then some geodesic in \(N_{i}\) is the core of an embedded tube of radius \(\log(3)/2\). As a corollary, the following is shown: Vol3 is the unique closed hyperbolic 3-manifold such that no closed geodesic is the core of an embedded tube of radius \(\log(3)/2\). The results above are proved by a rigorous computer-assisted procedure.