Statistics and compression of scl (joint with J. Maher)

In a hyperbolic group, a random word of length $n$ in the commutator subgroup has stable commutator length of order $n/\log{n}$. In any finitely generated group, either stable commutator length vanishes identically, or the result of a random walk of length $n$ conditioned to lie in the commutator subgroup has stable commutator length bounded above by order $n/\log{n}$ and below by order $\sqrt{n}$. The upper bounds are obtained by explicit estimates on random words and random geodesics in
free and hyperbolic groups. The lower bounds are obtained from properties of the statistical distribution of values of quasimorphisms. One result we prove of independent interest is a central limit theorem for values of the rotation quasimorphism on random walks in semigroups of homeomorphisms of the circle.