The realization problem for Jørgensen numbers

Yamashita, Yasushi; Yamazaki, Ryosuke

doi:10.1090/ecgd/331

The realization problem for Jørgensen numbers
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by Yasushi Yamashita and Ryosuke Yamazaki

Conform. Geom. Dyn. 23 (2019), 17-31

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