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Conformal Geometry and Dynamics

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The realization problem for Jørgensen numbers


Authors: Yasushi Yamashita and Ryosuke Yamazaki
Journal: Conform. Geom. Dyn. 23 (2019), 17-31
MSC (2010): Primary 30F40, 57M50
DOI: https://doi.org/10.1090/ecgd/331
Published electronically: February 25, 2019
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Abstract: Let $ G$ be a two-generator subgroup of $ \mathrm {PSL}(2, \mathbb{C})$. The Jørgensen number $ J(G)$ of $ G$ is defined by

$\displaystyle J(G) = \inf \{ \vert\mathrm {tr}^2 A-4\vert + \vert\mathrm {tr} [A,B]-2\vert \: ; \: G=\langle A, B\rangle \}. $

If $ G$ is a non-elementary Kleinian group, then $ J(G)\geq 1$. This inequality is called Jørgensen's inequality. In this paper, we show that, for any $ r\geq 1$, there exists a non-elementary Kleinian group whose Jørgensen number is equal to $ r$. This answers a question posed by Oichi and Sato. We also present our computer generated picture which estimates Jørgensen numbers from above in the diagonal slice of Schottky space.

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Additional Information

Yasushi Yamashita
Affiliation: Nara Women’s University, Kitauoyanishi-machi, Nara-shi, Nara 630-8506, Japan
Email: yamasita@ics.nara-wu.ac.jp

Ryosuke Yamazaki
Affiliation: Gakushuin Boys’ Senior High School, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-0031, Japan
Email: rsk.yamazaki.ms@gmail.com

DOI: https://doi.org/10.1090/ecgd/331
Keywords: J{\o}rgensen's inequality, J{\o}rgensen number, Kleinian groups
Received by editor(s): August 21, 2017
Received by editor(s) in revised form: April 15, 2018, and September 26, 2018
Published electronically: February 25, 2019
Additional Notes: This work was supported by JSPS KAKENHI Grant Number 26400088.
Article copyright: © Copyright 2019 American Mathematical Society