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 Discreteness by use of a test map Bull. Korean Math. Soc. 2012 Vol. 49, No. 1, 57-61 https://doi.org/10.4134/BKMS.2012.49.1.57Published online January 1, 2012 Liulan Li and Xi Fu Hengyang Normal University, Hunan Normal University Abstract : It is well known that one could use a fixed loxodromic or parabolic element of a non-elementary group $G\subset M(\overline{\mathbb R}^n)$ as a test map to test the discreteness of $G$. In this paper, we show that a test map need not be in $G$. We also construct an example to show that the similar result using an elliptic element as a test map does not hold. Keywords : discreteness, non-elementary M\"obius group, test map MSC numbers : Primary 30F40; Secondary 20H10 Full-Text :