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I was wondering if anyone knew a 'simple' proof of the fact that a pseudo-Anosov diffeomorphism of a closed surface $\Sigma$ is not reducible, in the sense that it does not fix the free homotopy class of a multi-curve on $\Sigma$?

Thanks for your attention!

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    $\begingroup$ One hint is that a pseudo-Anosov map (it is not actually a diffeomorphism) sends geodesics (in the singular CAT(0) metric) to (unparameterized) geodesics, and one can see therefore that the only possible fixed curves would have to be horizontal or vertical. But the leaves of the stable/unstable foliation are not closed (e.g. they get contracted/expanded by the mapping). $\endgroup$
    – Ian Agol
    Commented Dec 7, 2016 at 17:45
  • $\begingroup$ Well, I understand that a pseudo-Anosov map cannot fix a curve, my question is about the homotopy class of a curve. In which case it is not obvious to me that your argument concludes. $\endgroup$
    – Selim G
    Commented Dec 7, 2016 at 17:48
  • $\begingroup$ In each homotopy class of a locally CAT(0) metric, there is a unique (locally) geodesic representative. So a curve is preserved iff a geodesic is. I'm not sure that your question is appropriate for mathoverflow, which is aimed towards research-level questions. $\endgroup$
    – Ian Agol
    Commented Dec 7, 2016 at 17:53
  • $\begingroup$ Yes, but couldn't it be possible that a free homotopy class would be preserved without any of its representative being preserved? $\endgroup$
    – Selim G
    Commented Dec 7, 2016 at 17:56
  • $\begingroup$ That's not possible, because the function which maps a free homotopy class $C$ to its geodesic representative $\gamma(C)$ is a bijection which is equivariant with respect to the mapping class group, meaning that $\phi(\gamma(C)) = \gamma(\phi(C))$ for each $\phi \in \text{MCG}(\Sigma)$. So $C$ is fixed by $\phi$ if and only if $\gamma(C)$ is fixed by $\phi$. $\endgroup$
    – Lee Mosher
    Commented Dec 12, 2016 at 0:49

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Here are some of the details for the answers given by Ian and Lee in the comments.

Suppose that $S$ is the given (closed, connected, oriented) surface. Suppose that $f : S \to S$ is the given pseudo-Anosov map. By the definition of pseudo-Anosov, there is a pair of transverse, transversely measured foliations $F^+$ and $F^-$ in $S$ which are both preserved by $f$. Also, $f$ expands the transverse measure of one and contracts the transverse measure of the other, both by the same factor $\lambda > 1$. There are different conventions about which foliation gets called "stable" and which gets expanded... I'll call $F^+$ the unstable foliation and "draw" it as horizontal.

Suppose that $\gamma$ is a small arc in $S$, transverse to both foliations. Then we can "integrate" $\gamma$ against $F^\pm$ and get a pair of real numbers. I'll call these the "imaginary" and "real" periods of $\gamma$. This gives $S$ a locally euclidean metric, away from the common singularities of $F^\pm$. It is an exercise to show that the cone angle at a $k$-prong singularity equals $k \cdot \pi$. (Recall that $S$ is closed.)

Thus the transversely measured foliations $F^\pm$ gives us a "singular euclidean metric" on $S$. Let's call this metric $q = q_f$. This is the "induced geometric structure" we need. Note that $f$ does not preserve $q$. Instead (by our convention above) $f$ stretches the horizontal foliation (that is, $F^+$) and shrinks the vertical. That is, $f$ acts on $(S, q)$ as a piecewise affine map, non-affine only at the singularities, preserving area.

Suppose that $\alpha \subset S$ is a simple closed essential curve. We can "pull $\alpha$ tight" and obtain a $q$-geodesic $\alpha^*$. Note that $\alpha^*$ is typically unique - when it is not unique the union of all geodesic representatives of $\alpha$ foliate a "flat cylinder" $A(\alpha)$. Thus either $\alpha^*$ or $A(\alpha)$ is an invariant of the homotopy class of $\alpha$.

Finally, consider any curve $\alpha$. Set $\alpha_k = f^k(\alpha)$. We now compute the $q$-geodesic representatives $\alpha_k^*$. As Ian and Lee point out these tend (exponentially quickly in $k$) towards horizontal. Since $f$ does not preserve (any!) geodesic representatives, it does not preserve any homotopy classes.

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