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Two-generator free Kleinian groups and hyperbolic displacements

İlker S Yüce

Algebraic & Geometric Topology 14 (2014) 3141–3184

The log3 theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3–space 3 is moved a distance at least log3 by one of the noncommuting isometries ξ or η of 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 log3 theorem, the main result of this paper shows that every point in 3 is moved a distance at least log5 + 32 by one of the isometries ξη or ξη.

free Kleinian groups, hyperbolic displacements, $\log 3$ theorem
Mathematical Subject Classification 2010
Primary: 14E20, 54C40
Secondary: 46E25, 20C20
Received: 16 December 2009
Revised: 30 May 2014
Accepted: 30 May 2014
Published: 15 January 2015
İlker S Yüce
Basic Sciences Unit
TED University
Ziya Gökalp St.
No. 48, Kolej 06420
Çankaya, Ankara