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.)

are. $\endgroup$ – Sebastian Apr 9 '13 at 20:24any$S_{p,g}$. $\endgroup$ – HJRW Apr 13 '13 at 20:08