• Journal of the Korean Mathematical Society
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• v.44 no.3
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• pp.615-626
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• 2007

A pair of pants $\sum(0,\;3)$ is a building block of oriented surfaces. The purpose of this paper is to determine the discrete conditions for the holonomy group $\pi$ of hyperbolic structure of a pair of pants. For this goal, we classify the relations between the locations of principal lines and entries of hyperbolic matrices in $\mathbf{PSL}(2,\;\mathbb{R})$. In the level of the matrix group $\mathbf{SL}(2,\;\mathbb{R})$, we will show that the signs of traces of hyperbolic elements playa very important role to determine the discreteness of holonomy group of a pair of pants.

• The Pure and Applied Mathematics
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• v.17 no.4
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• pp.275-288
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• 2010

A one-holed torus ${\Sigma}$(l, 1) is a building block of oriented surfaces. In this paper we formulate the matrix presentations of the holonomy group of a one-holed torus ${\Sigma}$(1, 1) by the gluing method. And we present an algorithm for deciding the discreteness of the holonomy group of ${\Sigma}$(1, 1).

• Journal of the Korean Mathematical Society
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• v.51 no.2
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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.

• Journal of the Korean Mathematical Society
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• v.42 no.3
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• pp.555-571
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• 2005

A pair of pants $\Sigma(0,3)$ is a building block of oriented surfaces. The purpose of this paper is to formulate the matrix presentations of elements of the Teichmuller space of a pair of pants. In the level of the matrix group $SL(2,\mathbb{R})$, we shall show that an odd number of traces of matrix presentations of the generators of the fundamental group of $\Sigma(0,3)$ should be negative.

• The Pure and Applied Mathematics
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• v.11 no.1
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• pp.73-88
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• 2004

A punctured torus $\Sigma(1,1)$ is a building block of oriented surfaces. The goal of this paper is to formulate the matrix presentations of elements of the Teichmuller space of a punctured torus. Let $\cal{C}$ be a matrix presentation of the boundary component of $\Sigma(1,1)$.In the level of the matrix group $\mathbb{SL}$($\mathbb2,R$) we shall show that the trace of $\cal{C}$ is always negative.