# Two-Generator Free Kleinian Groups and Hyperbolic Displacements - Mathematics > Geometric Topology

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Abstract: The $\log 3$ Theorem, proved by Culler and Shalen, states that every point inthe hyperbolic 3-space is moved a distance at least $\log 3$ by one of thenon-commuting isometries $\xi$ or $\eta$ provided that $\xi$ and $\eta$generate a torsion-free, discrete group which is not co-compact and contains noparabolic. This theorem lies in the foundation of many techniques that providelower estimates for the volumes of orientable, closed hyperbolic 3-manifoldswhose fundamental group has no 2-generator subgroup of finite index and, as aconsequence, gives insights into the topological properties of these manifolds.In this paper, we show that every point in the hyperbolic 3-space is moved adistance at least $1-2\log5+3\sqrt{2}$ by one of the isometries in$\{\xi,\eta,\xi\eta\}$ when $\xi$ and $\eta$ satisfy the conditions given inthe $\log 3$ Theorem.

Autor: İlker S. Yüce

Fuente: https://arxiv.org/