Two-Generator Free Kleinian Groups and Hyperbolic Displacements

TL;DR

This paper extends the $ ext{log } 3$ Theorem for hyperbolic 3-space, demonstrating that certain two-generator groups move points at least a specific distance, which aids in understanding hyperbolic 3-manifold properties.

Contribution

It introduces a new displacement bound involving three isometries for two-generator Kleinian groups under conditions similar to the $ ext{log } 3$ Theorem, enhancing geometric estimates.

Findings

Every point in hyperbolic 3-space is moved at least $(1/2) ext{log}(5+3 ext{sqrt}(2))$ by one of the isometries in $\\{\xi,\,\eta,\,\xi\,\eta\\}$.

The result generalizes the $ ext{log } 3$ Theorem to include the product of generators.

Provides new lower bounds for displacement in hyperbolic geometry.

Abstract

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.