Abstract
We prove a rigidity theorem for the geometry of the unit ball in random subspaces of the norm in of a free group. In a free group of rank , a random word of length (conditioned to lie in ) has with high probability, and the unit ball in a subspace spanned by random words of length is 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.
Citation
Danny Calegari. Alden Walker. "Random rigidity in the free group." Geom. Topol. 17 (3) 1707 - 1744, 2013. https://doi.org/10.2140/gt.2013.17.1707
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