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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?

user7090
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  • Hint: Pair of pants has 3 boundary components. – Moishe Kohan May 06 '21 at 22:57
  • I thought a pair of pants has $a $boundaries and $b$ punctures such that $a+b=3$. – user7090 May 07 '21 at 01:30
  • One thing you should clear up: What do you mean by a genus $g$ surface with $k$ holes? Do you mean it has $k$ boundaries? Or $k$ punctures? Or either or both or some mix? – Lee Mosher May 08 '21 at 03:48
  • That is also not clear to me. The note states genus g with k-holes, there is no further explanation. With that said, when is it appropriate to consider both boundaries and punctures on the same footing? – user7090 May 08 '21 at 12:01

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