Tubes in hyperbolic 3-manifolds (Q1873295)

The author proves a lower bound on the volume of a maximal precisely invariant tube of radius \(r\) which meets itself at an angle \(\theta\) in a hyperbolic 3-orbifold. This result implies that any closed orientable hyperbolic 3-manifold has volume at least \(0.276666\) (recall that the smallest known volume manifold, obtained by Weeks, by Fomanko and Matveev and by Przytycki, has volume \(0.9427\ldots\)). Also, lower bounds on the volumes of closed hyperbolic 3-manifolds which have symmetries of large order are obtained. In particular, it is shown that the order of the symmetry group of the smallest volume hyperbolic 3-manifold is of the form \(2^m 3^n\) for some \(m, n \geq 0\) (recall that the symmetry group of the smallest known volume manifold from above is the dihedral group of order 12).