Discreteness by use of a test map

Liulan Li; Xi Fu

doi:10.4134/BKMS.2012.49.1.57

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2012; 49(1): 57-61

Printed January 1, 2012

https://doi.org/10.4134/BKMS.2012.49.1.57

Discreteness by use of a test map

Liulan Li and Xi Fu

Hengyang Normal University, Hunan Normal University

Abstract

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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