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Good evening everyone,

an outer automorphism $[\phi]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism $h\colon M\stackrel{\cong}{\to}M$, where $M$ is a compact surface with nonempty boundary.

Bestvina and Handel gave a classification of those outer automorphisms that are induced by a pseudo-Anosov homeomorphism of a compact surface with connected boundary ('Train tracks and automorphisms of free groups', Annals, 1992):

Theorem: $[\phi]\in Out(F_n)$ is induced by a pseudo-Anosov homeomorphism of a compact surface with one boundary component if and only if each $[\phi]^l$ is irreducible and there is a conjugacy class $s\in \mathcal{C}(F_n)$ such that $[\phi] (s)=s$ or $[\phi] (s)=\overline{s}$.

Can we deduce from this result a complete classification of all geometric outer automorphisms of $F_n$? Or do their assumptions on the surface and the homeomorphism not allow for a corollary that treats the general case? (We obviously can't just drop 'pseudo-Anosov' and 'one boundary component' from the statement of the theorem.)

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  • $\begingroup$ Are you talking about Thurston's classification of surface diffeomorphisms? If so, yes, you just paste together the relationship between $Out(F_n)$ and mapping class groups, and Thurston's classification. The way you've written your question it's not clear if you have this in mind: en.wikipedia.org/wiki/Nielsen%E2%80%93Thurston_classification $\endgroup$
    – Ryan Budney
    Commented Apr 9, 2013 at 18:29
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    $\begingroup$ Every homeomorphism of a surface $M$ with nonempty boundary induces an outer automorphisms of the fundamental group of $M$, which is free. However, not every outer automorphism of a free group is induced by a surface homeomorphism this way. My question is whether we understand all outer automorphisms of free groups that are. $\endgroup$
    – Sebastian
    Commented Apr 9, 2013 at 20:24
  • $\begingroup$ The following theorem might give you a partial answer. This is from the book "A primer on mapping class group." by Farb and Margalit. $\mathbf{Theorem:}$ Let $S_{g,p}$ be the surface of genus with p puncture. Let $Out^*(\pi_1(S_{g,p}))$ is the subgroup of $Out(\pi_1(S_{g,p}))$ consisting of those elements which preserves the set of conjugacy classes of simple closed curves surrounding individual punctures. Then There is an isomorphism between $Mod^{\pm}(S_{p,g})$ and $Out^*(\pi_1(S_{g,p}))$. $\endgroup$
    – Cusp
    Commented Apr 10, 2013 at 5:14
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    $\begingroup$ Cusp - that theorem characterizes when $\phi$ is a geometric automorphism of some given $S_{p,g}$. The theorem of Bestvina--Handel characterizes exactly when $\phi$ is a pseudo-Anosov automorphism of any $S_{p,g}$. $\endgroup$
    – HJRW
    Commented Apr 13, 2013 at 20:08
  • $\begingroup$ Sebastian - this is a good question, which also occurred to me when reading Bestvina--Handel's paper. I don't see how the general case would follow from their theorem, but it seems conceivable that their techniques could handle the general case. $\endgroup$
    – HJRW
    Commented Apr 13, 2013 at 20:13

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I think the short answer is "No", you cannot deduce from this result a classification of all geometric outer automorphisms.

I think it might eventually be possible to obtain a classification of geometric outer automorphisms from the more powerful "relative train track / lamination" machinery developed in the works of Bestvina, Feighn, and Handel, although that has not been done. Nonetheless one can deduce bits and pieces of such a classification.

For instance, suppose that $\phi \in Out(F_n)$ has an attracting lamination $\Lambda$ with the property that the smallest free factor of $F_n$ that supports $\Lambda$ is the whole free group. In this case one can prove that $\phi$ is geometric if and only if there exists a finite $\phi$-invariant set of root-free conjugacy classes $c_1,...,c_k$ such that the smallest free factor of $F_n$ that supports $c_1,...,c_k$ is also the whole free group, $c_1,...,c_k$ are the only root-free conjugacy classes that are not attracted to $\Lambda$ under iteration of $\phi$, and a few other nondegeneracy conditions hold. The picture to keep in mind is that $c_1,...,c_k$ represent the boundary components of a surface on which $\phi$ is represented as a pseudo-Anosov homoemorphism with unstable lamination $\Lambda$. The "nondegeneracy" conditions I mentioned are needed to avoid counterexamples where, say, three of the $c's$ are identified to the same closed curve, and these conditions can be expressed in an intrinsic manner in terms of ``Nielsen theory'' which means the asymptotic behavior of automorphisms representing the outer automorphism $\phi$.

This statement can be found in a slightly different form in Proposition 2.38 of the paper Subgroup classification in $Out(F_n)$ by Handel and myself, and in this exact form in the soon-to-appear Part III of the expanded version "Subgroup decomposition in $Out(F_n)$".

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