Two-generator free Kleinian groups and hyperbolic displacements

Abstract

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 $\xi$ or $\eta$ of ${ℍ}^3$ provided that $\xi$ and $\eta$ 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 $log\sqrt{5+3\sqrt{2}}$ by one of the isometries $\xi$, $\eta$ or $\xi \eta$.