Mathematics – Geometric Topology
Scientific paper
2009-11-25
Mathematics
Geometric Topology
85 Pages. Format is updated. A number of typos are corrected. The use of parameterizations made precise. Lemma 1.1 is updated
Scientific paper
The $\log 3$ Theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3-space is moved a distance at least $\log 3$ by one of the non-commuting isometries $\xi$ or $\eta$ provided that $\xi$ and $\eta$ generate a torsion-free, discrete group which is not co-compact and contains no parabolic. This theorem lies in the foundation of many techniques that provide lower estimates for the volumes of orientable, closed hyperbolic 3-manifolds whose fundamental group has no 2-generator subgroup of finite index and, as a consequence, gives insights into the topological properties of these manifolds. In this paper, we show that every point in the hyperbolic 3-space is moved a distance at least $(1/2)\log(5+3\sqrt{2})$ by one of the isometries in $\{\xi,\eta,\xi\eta\}$ when $\xi$ and $\eta$ satisfy the conditions given in the $\log 3$ Theorem.
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