Let G be a two generator subgroup of PSL(2, C). The Jørgensen number J(G) of G is defined byIf G is a non-elementary Kleinian group, then J(G) ≥ 1. This inequality is called Jørgensen's inequality. In this paper, we show that, for any r ≥ 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.Oichi-Sato [16] claimed that if r = 1, 2, 3 or r ≥ 4, then there is a non-elementary Kleinian group whose Jørgensen number is equal to r. (See also [3,23,24].) Though Jørgensen's inequality is sharp, constructing a Kleinian group with small Jørgensen 2010 Mathematics Subject Classification. 30F40, 57M50.
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