Random rigidity in the free group

Calegari, Danny; Walker, Alden

doi:10.2140/gt.2013.17.1707

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Random rigidity in the free group

Danny Calegari and Alden Walker

Geometry & Topology 17 (2013) 1707–1744

DOI: 10.2140/gt.2013.17.1707

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 $scl (w) = log (2 k - 1) n ∕ 6 log (n) + o (n ∕ log (n))$ 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.

Dedicated to the memory of Andrew Lange

Keywords

Gromov norm, stable commutator length, symbolic dynamics, rigidity, law of large numbers

Mathematical Subject Classification 2010

Primary: 20P05, 20F67, 57M07

Secondary: 20F65, 20J05

References

Publication

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

Authors

Danny Calegari
Department of Mathematics University of Chicago 5734 S University Avenue Chicago, IL 60637 USA
http://math.uchicago.edu/~dannyc
Alden Walker
Department of Mathematics University of Chicago 5734 S University Avenue Chicago, IL 60637 USA
http://math.uchicago.edu/~akwalker

Volume 17, issue 3 (2013)