Geometry & Topology

Random rigidity in the free group

Danny Calegari and Alden Walker

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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(2k1)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.

Article information

Geom. Topol., Volume 17, Number 3 (2013), 1707-1744.

Received: 29 June 2011
Revised: 5 October 2012
Accepted: 27 March 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20P05: Probabilistic methods in group theory [See also 60Bxx] 20F67: Hyperbolic groups and nonpositively curved groups 57M07: Topological methods in group theory
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20J05: Homological methods in group theory

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


Calegari, Danny; Walker, Alden. Random rigidity in the free group. Geom. Topol. 17 (2013), no. 3, 1707--1744. doi:10.2140/gt.2013.17.1707.

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