Singularity properties of random free group automorphisms and of random trees in the boundary of Outer space

Abstract: It is known that, under mild assumptions, for a free group $F_r$ of finite rank $r>2$, a "random" element $\phi_n\in Out(F_r)$, obtained after $n$ steps of a random walk on $Out(F_r)$, is fully irreducible (a free group analog of being pseudoAnosov), and that an a.e. trajectory of the way converges to a point in the boundary of the CullerVogtmann Outer space $CV_r$. We prove that generically the attracting $\mathbb R$-tree $T_+(\phi_n)\in \partial CV_r$ for such a random fully irreducible $\phi_n$ is trivalent (that is, all branch points of $T_+$ have degree 3) and nongeometric, (that is $T_+$ is not the dual tree of any measured foliation of a finite 2-complex).

Similarly, for the exit/harmonic measure $\nu$ of the random walk on the boundary $\partial CV_r$ of the Outer space, we prove that a $\nu$-a.e. $\mathbb R$-tree $T\in \partial CV_r$ is trivalent and nongeometric. The talk is based on joint work with Joseph Maher, Catherine Pfaff and Samuel Taylor.

group theory

Audience: researchers in the topic

Séminaire de groupes et géométrie