###
Index properties of random automorphisms of free groups

####
Ilya Kapovich (CUNY)

Abstract: For automorphisms of the free group $F_r$, being ``fully irreducible” is the main analog of the property of being a pseudo-Anosov element of the mapping class group.
It has been known, because of general results about random walks on groups acting on Gromov-hyperbolic spaces, that a ``random” (in the sense of being generated by a long random walk) element $\phi$ of $Out(F_r)$ is fully irreducible and atoroidal. But finer structural properties of such random fully irreducibles $\phi\in Out(F_r)$ have not been understood.
We prove that for a ``random” $\phi\in Out(F_r)$ (where $r\ge 3$), the attracting and repelling $\mathbb R$-trees of $\phi$ are trivalent, that is all of their branch points have valency three, and that these trees are non-geometric (and thus have index $<2r-2$). The talk is based on a joint paper with Joseph Maher, Samuel Taylor and Catherine Pfaff.