Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2007; 44(3): 615-626

Printed May 1, 2007

Copyright © The Korean Mathematical Society.

Discrete conditions for the holonomy group of a pair of pants

Hong Chan Kim

Korea University

Abstract

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