Journal of the
Korean Mathematical Society JKMS

A pair of pants $\Sigma(0,3)$ is a building block of oriented surfaces. The purpose of this paper is to formulate the matrix presentations of elements of the Teichm\"uller space of a pair of pants. In the level of the matrix group ${\bf SL}(2,\mathbb R),$ we shall show that an odd number of traces of matrix presentations of the generators of the fundamental group of $\Sigma(0,3)$ should be negative.

Keywords: a pair of pants, hyperbolic structure, Teichm\"uller space, holonomy homomorphism, discrete group

MSC numbers: 32G15, 57M50