Random rigidity in the free group

TL;DR

This paper establishes a rigidity property of the scl norm in free groups, showing that the geometry of the unit ball in random subspaces approximates an octahedron, with implications for hyperbolic groups and manifolds.

Contribution

It proves a new rigidity theorem for the scl norm in free groups and suggests a potential extension to hyperbolic groups and manifolds.

Findings

Random words have scl proportional to their length with high probability.

The unit ball in subspaces spanned by random words approximates an octahedron.

Potential to recover geodesic lengths from bounded cohomology in hyperbolic manifolds.

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 [F,F]) has scl(w)=log(2k-1)n/6log(n) + o(n/log(n)) with high probability, and the unit ball in a subspace spanned by d random words of length O(n) 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.