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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

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. In particular, we show that the commutator subgroup [B,B] admits a stably unbounded conjugation invariant norm. This answers an open problem posed by Burago, Ivanov and Polterovich.

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