  We define a quasihomomorphism from braid groups to the concordance group of
knots and examine its properties and consequences of its existence. In
particular, we provide a relation between the stable four ball genus in the
concordance group and the stable commutator length in braid groups, and produce
examples of infinite families of concordance classes of knots with uniformly
bounded four ball genus. We also provide applications to the geometry of the
infinite braid group. In particular, we show that its commutator subgroup
admits a stably unbounded conjugation invariant norm. This answers an open
problem posed by Burago, Ivanov and Polterovich.
