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(2k1) 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.
This talk is part of the Probability series.
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