Two-generator free Kleinian groups and hyperbolic displacements

Yüce, İlker S

doi:10.2140/agt.2014.14.3141

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

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

DOI: 10.2140/agt.2014.14.3141

Abstract

The $log 3$ theorem, proved by Culler and Shalen, states that every point in the hyperbolic $3$ –space $ℍ^{3}$ is moved a distance at least $log 3$ by one of the noncommuting isometries $ξ$ or $η$ of $ℍ^{3}$ provided that $ξ$ and $η$ 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 $log 3$ 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 $ξ$ , $η$ or $ξ η$ .

Keywords

free Kleinian groups, hyperbolic displacements, $\log 3$ theorem

Mathematical Subject Classification 2010

Primary: 14E20, 54C40

Secondary: 46E25, 20C20

References

Publication

Received: 16 December 2009

Revised: 30 May 2014

Accepted: 30 May 2014

Published: 15 January 2015

Authors

İlker S Yüce
Basic Sciences Unit TED University Ziya Gökalp St. No. 48, Kolej 06420 Çankaya, Ankara Turkey

Volume 14, issue 6 (2014)