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Random rigidity in the free group

Danny Calegari and Alden Walker

Geometry & Topology 17 (2013) 1707–1744

We prove a rigidity theorem for the geometry of the unit ball in random subspaces of the scl norm in B1H 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)n6log(n) + o(nlog(n)) with high probability, and the unit ball in a subspace spanned by d random words of length O(n) is C0 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.

Dedicated to the memory of Andrew Lange

Gromov norm, stable commutator length, symbolic dynamics, rigidity, law of large numbers
Mathematical Subject Classification 2010
Primary: 20P05, 20F67, 57M07
Secondary: 20F65, 20J05
Received: 29 June 2011
Revised: 5 October 2012
Accepted: 27 March 2013
Published: 21 June 2013
Proposed: Jean-Pierre Otal
Seconded: Dmitri Burago, Leonid Polterovich
Danny Calegari
Department of Mathematics
University of Chicago
5734 S University Avenue
Chicago, IL 60637
Alden Walker
Department of Mathematics
University of Chicago
5734 S University Avenue
Chicago, IL 60637