Two-generator free Kleinian groups and hyperbolic displacements

Yüce, İlker S¹

¹Basic Sciences Unit, TED University, Ziya Gökalp St., No. 48, Kolej 06420, Çankaya, Ankara, Turkey

Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3141-3184

Résumé

The log3 theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3–space ℍ3 is moved a distance at least log3 by one of the noncommuting isometries ξ or η of ℍ3 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 ℍ3 is moved a distance at least log5 + 32 by one of the isometries ξ, η or ξη.

Détail

DOI : 10.2140/agt.2014.14.3141

Classification : 14E20, 54C40, 46E25, 20C20
Keywords: free Kleinian groups, hyperbolic displacements, $\log 3$ theorem

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