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Statistics and compression of scl. (English) Zbl 1351.37214
Summary: 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 non-degenerately 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 and Fujiwara and others on the infinite dimensionality of two-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 $$n$$.

##### MSC:
 37H10 Generation, random and stochastic difference and differential equations 57M60 Group actions on manifolds and cell complexes in low dimensions 20F12 Commutator calculus 68R15 Combinatorics on words 20F67 Hyperbolic groups and nonpositively curved groups 57M50 General geometric structures on low-dimensional manifolds 20P05 Probabilistic methods in group theory
##### Keywords:
random (geodesic) words; random walks; hyperbolic groups
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