Random rigidity in the free group

Abstract

We prove a rigidity theorem for the geometry of the unit ball in random subspaces of the $scl$ norm in $B_1^H$ of a free group. In a free group $F$ of rank $k$, a random word $w$ of length $n$ (conditioned to lie in $\left[F,F\right]$) has $scl\left(w\right)=log\left(2k-1\right)n∕6log\left(n\right)+o\left(n∕log\left(n\right)\right)$ with high probability, and the unit ball in a subspace spanned by $d$ random words of length $O\left(n\right)$ is $C^0$ close to a (suitably affinely scaled) octahedron.

A conjectural generalization to hyperbolic groups and manifolds (discussed in the appendix) would show that the length of a random geodesic in a hyperbolic manifold can be recovered from the bounded cohomology of the fundamental group.