Two-Generator Free Kleinian Groups and Hyperbolic Displacements

The log3 theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3‐space H 3 is moved a distance at least log3 by one of the noncommuting isometries or of H 3 provided that and generate a torsion-free, discrete group which is not cocompact and contains no parabolic. This theorem lies in the foundations of many techniques that provide lower estimates for the volumes of orientable, closed hyperbolic 3‐manifolds whose fundamental groups have no 2‐ generator subgroup of finite index and, as a consequence, gives insights into the topological properties of these manifolds. Under the hypotheses of the log3 theorem, the main result of this paper shows that every point in H 3 is moved a distance at least log p 5C3 p 2 by one of the