Communications in Analysis and Geometry

Volume 26 (2018)

Number 6

Symmetric decompositions of free Kleinian groups and hyperbolic displacements

Pages: 1375 – 1447



İlker S. Yüce (Department of Mathematics, Faculty of Arts and Sciences, Yeditepe University, Istanbul, Turkey)


In this paper, it is shown that every point in the hyperbolic $3$-space is moved at a distance at least $\frac{1}{2} \log (12 \cdot 3^{k-1}-3)$ 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.

Received 11 January 2016

Published 29 March 2019