Abstract：It is known that, under mild assumptions, for a free group $F$ of finite rank $r>2$, a "random" element $\phi_n\in Out(F)$ obtained after $n$ steps of a random walk on $Out(F)$ is fully irreducible (a free group analog of being pseudo-Anosov), and that an a.e. trajectory of the way converges to a point in the boundary of the Culler-Vogtmann Outer space $CV(F)$. We prove that generically the attracting $\mathbb R$-tree $T_+(\phi_n)$ for such a random fully irreducible $\phi_n$ is trivalent (that is, all branch points of $T_+$ have degree 3) and non-geometric, that is $T_+$ is not the dual tree of any measured foliation of a finite 2-complex. Similarly, for the exit/harmonic measure of the random walk on the boundary $\partial CV(F)$ of the Outer space, we prove that a generic $\mathbb R$-tree $T\in \partial CV(F)$ is trivalent and non-geometric.
The talk is based on joint walk with Joseph Maher, Catherine Pfaff and Samuel Taylor.
