The realization problem for Jørgensen numbers

Yasushi Yamashita; Ryosuke Yamazaki

doi:10.1090/ecgd/331

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arXiv · 1703.07732

The realization problem for Jørgensen numbers

Yasushi Yamashita·Ryosuke Yamazaki

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Abstract

Let G be a two generator subgroup of PSL(2,C). The Jorgensen number J(G) of G is defined by J(G)=inf{ |tr^2 A-4|+|tr[A,B]-2| ; G= }. If G is a non-elementary Kleinian group, then J(G) >= 1. This inequality is called Jorgensen's inequality. In this paper, we show that, for any r >= 1, there exists a non-elementary Kleinian group whose Jorgensen number is equal to r. This answers a question posed by Oichi and Sato. We also present our computer generated picture which estimates Jorgensen numbers from above in the diagonal slice of Schottky space.

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Yasushi Yamashita, Ryosuke Yamazaki. 2017-03-22. The realization problem for Jørgensen numbers. https://doi.org/10.1090/ecgd%2F331

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