Symmetric Decompositions of Free Kleinian Groups and Hyperbolic Displacements

TL;DR

This paper establishes lower bounds on hyperbolic displacements in free Kleinian groups, showing that points are moved at least a certain distance by short isometries, and discusses conjectures on displacement bounds for symmetric isometry subsets.

Contribution

It provides explicit lower bounds for hyperbolic displacements in torsion-free, non-co-compact free Kleinian groups and proposes conjectures on displacement maxima for symmetric isometry subsets.

Findings

Every point is moved at least 0.5 log(12 * 3^{k-1} - 3) by some isometry of length ≤ k.

Lower bounds for hyperbolic displacements are established for certain Kleinian groups.

Conjectures are proposed for maximum displacements in symmetric subsets of isometries.

Abstract

In this paper, it is shown that every point in the hyperbolic 3-space is moved at a distance at least $0.5\log\left(12\cdot 3^{k-1}-3\right)$ by one of the isometries of length at most $k\geq 2$ in a 2-generator Klenian group $\Gamma$ which is torsion-free, not co-compact and contains no parabolic. Also some lower bounds for the maximum of hyperbolic displacements given by symmetric subsets of isometries in purely loxodromic finitely generated free Kleinian groups are conjectured.