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Jørgensen’s inequality; quaternionic hyperbolic space; embedding.

Jørgensen’s inequality gives a necessary condition for a nonelementary two generator subgroup of $SL(2,\mathbb{C})$ to be discrete. By embedding $SL(2,\mathbb{C})$ into $Û(1,1;\mathbb{H})$, we obtain a new type of Jørgensen’s inequality, which is in terms of the coefficients of involved isometries. We provide an example to show that this result gives an improvement over the classical Jørgensen’s inequality.

Hua Wang¹ Yueping Jiang² Wensheng Cao³

1. Department of Mathematics, Changsha University of Science and Technology, Changsha, Hunan 410076, People’s Republic of China
2. Department of Applied Mathematics, Hunan University, Changsha 410082, People’s Republic of China
3. School of Mathematics and Computational Science, Wuyi University, Jiangmen, Guangdong 529020, People’s Republic of China

Manuscript received

12 January 2012

Manuscript revised

27 April 2012