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