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I am recently reading the paper "Natural triangulations associated to a surface" by B.H. Bowditch and D.B.A. Epstein (http://www.sciencedirect.com/science/article/pii/0040938388900080#), where they described a natural triangulation for moduli space of a surface. In the first part they produces a ideal triangulation (the spinal triangulation) of a closed surface of genus $g$ with at least one cusp (i.e. $S$ is a closed surface with some points $P$ such that $S-P$ has complete hyperbolic structure of finite area). In the second part they defined the notion of a surface with nodes (a node is just a compact subsurface with boundary). Now suppose we have a surface $S$ with nodes $N$, they have considered the components of $S-N$ (with at-least one cusp in each component) and applied the theorem from the first part to obtain a ideal triangulation.

My question is the component of $S-N$ are likely to be surface with boundary which is not considered in the first part, so how can we construct an ideal triangulation of a surface with boundary using the corresponding ideal triangulation in the closed case.

P.S: If I have missed something in that paper please let me know. Suggestions are welcome.

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up vote 5 down vote accepted

Since $N$ is compact, $S - N$ is not compact. In particular $S - N$ is homeomorphic to a punctured surface: here a punctured surface is a compact surface without boundary minus a finite set of points.

It may help to copy out their definitions word-by-word. Then find an example for each definition.

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Just to add emphasis: $S-N$ has no boundary, contrary to the OPs statement that the components of $S-N$ are likely to be surfaces with boundary. –  Lee Mosher Mar 17 '13 at 18:05
    
Thanks. I missed this point. –  Cusp Mar 17 '13 at 18:14

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