# Nielsen-Thurston classification of homeomorphisms for open surfaces?

In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $\ f:M \rightarrow M$ of an open surface $M$ which is a $k$-times punctured sphere with $k \geq 3$. Think about $M$ for example as an open annulus $\mathbb{R}/\mathbb{Z} \times (0,1)$ with $k-2$ points removed.

What confuses me is that I cannot seem to find any reference for the Thurston classification of homeomorphisms for open surfaces. Everywhere it's done only for compact surfaces w/o boundary. I have no idea why it should still work for the said punctured spheres. Any help is much appreciated!

He uses it to prove that for an area-preserving homeo $f:M \rightarrow M$ without periodic points, there has to be a power $f^n$ which is isotopic to the identity. Following the Thurston classification, there are essentially three possibilities for $f$:

1. $f$ is periodic (i.e. there is a power $f^n$ which is isotopic to id)
2. $f$ is pseudo-anosov or has pseudo-anosov components (which would imply periodic points)
3. $f$ is reducible only with periodic components (which also implies periodic points)

Thanks alot :)

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The theories for an open surface $S$ and for a compact surface with boundary $\overline S$ whose interior is identified with $S$ are the same. The inclusion of $S$ into $\overline S$ defines an isomorphism of mapping class groups. The case of a compact surface with boundary is covered in all of the early sources that I checked:

• Chapter 8 of the book of Fathi, Laudenbach, and Poenaru (translation by Kim and Margalit).
• The paper of Handel and Thurston, "New proofs of some results of Nielsen". Adv. in Math. 56 (1985), no. 2, 173–191.
• Section 9 of the paper of Richie Miller, "Geodesic laminations from Nielsen's viewpoint". Adv. in Math. 45 (1982), no. 2, 189–212.

I'll add that the mapping class group of $\overline S$, defined as homeomorphisms modulo isotopies, is not the same as the mapping class group of $\overline S$ relative to its boundary, defined as homeomorphisms fixing each boundary point modulo isotopies fixing each boundary point. So when you read about the mapping class group of a compact surface with boundary, you have to read carefully to see which of the two definitions is intended. The relative mapping class group is also covered by Thurston's classification theory, and it is only a minor extension requiring that one pay attention to Dehn twists about boundary circles. However it is indeed a little harder to find the correct statements in the literature.

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See Theorem 13.2 in Farb and Margalit's primer on Mapping class groups.

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right, thanks! I'll check. – eddard Apr 7 '14 at 14:00