J. Korean Math. Soc. 2007; 44(3): 615-626
Printed May 1, 2007
Copyright © The Korean Mathematical Society.
Hong Chan Kim
Korea University
A pair of pants $\Sigma(0,3)$ 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 ${\bf PSL}(2,\mathbb R)$. In the level of the matrix group ${\bf SL}(2,\mathbb R),$ we will show that the signs of traces of hyperbolic elements play a 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
MSC numbers: 32G15, 57M50
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