Statistics and compression of scl

Abstract

We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting nondegenerately on hyperbolic spaces. In either case, we show that with high probability stable commutator length of an element of length \(n\) is of order \(n/\log{n}\). This establishes quantitative refinements of qualitative results of Bestvina-Fujiwara and others on the infinite dimensionality of 2-dimensional bounded cohomology in groups acting suitably on hyperbolic spaces, in the sense that we can control the geometry of the unit balls in these normed vector spaces (or rather, in random subspaces of their normed duals). As a corollary of our methods, we show that an element obtained by random walk of length \(n\) in a mapping class group cannot be written as a product of fewer than \(O(n/\log{n})\) reducible elements, with probability going to 1 as \(n\) goes to infinity. We also show that the translation length on the complex of free factors of a random walk of length \(n\) on the outer automorphism group of a free group grows linearly in \(n\).