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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
MathSciNet review: 3916474
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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 \[ J(G) = \inf \{ |\mathrm {tr}^2 A-4| + |\mathrm {tr} [A,B]-2| \: ; \: G=\langle A, B\rangle \}. \] 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
MR Author ID: 310816
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

Keywords: Jørgensen’s inequality, Jørgensen 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