#### Volume 15, issue 5 (2015)

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Concordance group and stable commutator length in braid groups

### Michael Brandenbursky and Jarek Kędra

Algebraic & Geometric Topology 15 (2015) 2859–2884
##### 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.

##### Keywords
braid group, concordance group, quasimorphism, conjugation invariant norm, commutator length, four ball genus
##### Mathematical Subject Classification 2010
Primary: 20F36, 57M25
Secondary: 20F69
##### Publication
Received: 21 August 2014
Revised: 9 February 2015
Accepted: 13 February 2015
Published: 10 December 2015
##### Authors
 Michael Brandenbursky Department of Mathematics Ben-Gurion University Beer-Sheva Israel Jarek Kędra Institute of Mathematics University of Aberdeen Aberdeen AB243UE UK Instytut Matematyki Uniwersytet Szczeciǹski 70-451 Szczecin Poland