I am reading a set of notes on Representations of Surface Groups made for a workshop at AIM.
It states that a genus g surface with $k$ holes has a decomposition (I am assuming this is referring to that surface's fundamental domain) as a $4g+k$-gon with $2g$-side identifications.
As far as I understand, I can treat a pair of pants as having three-holes, in which case its fundamental domain should be a $4(0) +3=3$-gon, i.e. a triangle.
However, the fundamental domain of a pair of pants (from what I have seen) is a hexagon and not a triangle.
What is going wrong?