# Classification of geometric outer automorphisms of free groups

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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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 –  Ryan Budney Apr 9 '13 at 18:29
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. –  Sebastian Apr 9 '13 at 20:24
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 $Out^*(\pi_1(S_{g,p}))$. –  Cusp Apr 10 '13 at 5:14
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 $S_{p,g}$. –  HJRW Apr 13 '13 at 20:08
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. –  HJRW Apr 13 '13 at 20:13

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 $Out(F_n)$".