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A pair of pants Σ(0, 3) is a building block of oriented surfaces. The purpose of this paper is to determine the discrete conditions for the holonomy group π 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 PSL(2,R). In the level of the matrix group SL(2,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.

DISCRETE CONDITIONS FOR THE HOLONOMY GROUP OF A PAIR OF PANTS