DISCRETE CONDITIONS FOR THE HOLONOMY GROUP OF A PAIR OF PANTS

Abstract

A pair of pants <TEX>$\sum(0,\;3)$</TEX> is a building block of oriented surfaces. The purpose of this paper is to determine the discrete conditions for the holonomy group <TEX>$\pi$</TEX> 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 <TEX>$\mathbf{PSL}(2,\;\mathbb{R})$</TEX>. In the level of the matrix group <TEX>$\mathbf{SL}(2,\;\mathbb{R})$</TEX>, 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.