#### Volume 14, issue 6 (2014)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Two-generator free Kleinian groups and hyperbolic displacements

### İlker S Yüce

Algebraic & Geometric Topology 14 (2014) 3141–3184
##### 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 $\xi \eta$.

##### Keywords
free Kleinian groups, hyperbolic displacements, $\log 3$ theorem
##### Mathematical Subject Classification 2010
Primary: 14E20, 54C40
Secondary: 46E25, 20C20