• 대한수학회보
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• 제50권3호
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• pp.747-752
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• 2013

It's known that one could use a fixed loxodromic or parabolic element in $M(\bar{\mathbb{R}}^n)$ as a test map to test the discreteness of a non-elementary M$\ddot{o}$bius group G. In this paper, we discuss the discreteness of G by using a fixed elliptic element.

• 대한수학회지
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• 제51권2호
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• pp.403-426
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• 2014

The purpose of this paper is to find expressions of the Fricke spaces of some basic surfaces which are a three-holed sphere ${\sum}$(0, 3), a one-holed torus ${\sum}$(1, 1), and a four-holed sphere ${\sum}$(0, 4). For this goal, we define the involutions corresponding to oriented axes of loxodromic elements and an inner product <,> which gives the information about locations of axes of loxodromic elements. The signs of traces of holonomy elements, which are calculated by lifting a representation from PSL(2, $\mathbb{C}$) to SL(2, $\mathbb{C}$), play a very important role in determining the discreteness of holonomy groups.