# DISCRETE CONDITIONS FOR THE HOLONOMY GROUP OF A PAIR OF PANTS

• Kim, Hong-Chan (DEPARTMENT OF MATHEMATICS EDUCATION KOREA UNIVERSITY)
• Published : 2007.05.31
• 86 9

#### Abstract

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.

#### Keywords

a pair of pants;hyperbolic structure;hyperbolic matrix;discrete holonomy group

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