Concordance group and stable commutator length in braid groups

Abstract

We define quasihomomorphisms from braid groups to the concordance group of knots and examine their properties and consequences of their 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 $B_{\infty }$. In particular, we show that the commutator subgroup $\left[B_{\infty },B_{\infty }\right]$ admits a stably unbounded conjugation invariant norm. This answers an open problem posed by Burago, Ivanov and Polterovich.