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Random rigidity in the free group

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If G is a group, and [G,G] is its commutator subgroup, the commutator length of an element w (denoted cl(w)) is the least number of commutators in G whose product is w; and the stable commutator length scl(w) is the limit of cl(w^n)/n as n goes to infinity. Stable commutator length is related to bounded cohomology and quasimorphisms, but is notoriously difficult to calculate exactly, or even to approximate. However, we show that 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(2k-1) n / 6 log(n) + o(n / log(n)) with high probability. The proof combines elements from ergodic theory and combinatorics. This is joint work with Alden Walker.

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