Geometry & Topology
- Geom. Topol.
- Volume 17, Number 3 (2013), 1707-1744.
Random rigidity in the free group
We prove a rigidity theorem for the geometry of the unit ball in random subspaces of the norm in of a free group. In a free group of rank , a random word of length (conditioned to lie in ) has with high probability, and the unit ball in a subspace spanned by random words of length is 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.
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.gt/1513732616