The realization problem for Jørgensen numbers

Yamashita, Yasushi; Yamazaki, Ryosuke

doi:10.1090/ecgd/331

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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 be a two-generator subgroup of $\mathrm {PSL}(2, \mathbb{C})$ . The Jørgensen number of is defined by

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

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

. 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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