Volume 17, issue 3 (2013)

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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

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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