Geodesic
Concordance group and stable commutator length in braid groups
Brandenbursky, Michael1 ; Kędra, Jarek2
1Department of Mathematics, Ben-Gurion University, Beer-Sheva, Israel
2Institute of Mathematics, University of Aberdeen, Aberdeen AB243UE, UK, Instytut Matematyki, Uniwersytet Szczeciǹski, 70-451 Szczecin, Poland
Algebraic and Geometric Topology, Tome 15 (2015) no. 5, pp. 2859-2884
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

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.

DOI : 10.2140/agt.2015.15.2861
Classification : 20F36, 57M25, 20F69
Keywords: braid group, concordance group, quasimorphism, conjugation invariant norm, commutator length, four ball genus

Brandenbursky, Michael  1   ; Kędra, Jarek  2

1 Department of Mathematics, Ben-Gurion University, Beer-Sheva, Israel
2 Institute of Mathematics, University of Aberdeen, Aberdeen AB243UE, UK, Instytut Matematyki, Uniwersytet Szczeciǹski, 70-451 Szczecin, Poland 
@article{10_2140_agt_2015_15_2861,
     author = {Brandenbursky, Michael and K\k{e}dra, Jarek},
     title = {Concordance group and stable commutator length in braid groups},
     journal = {Algebraic and Geometric Topology},
     pages = {2859--2884},
     year = {2015},
     volume = {15},
     number = {5},
     doi = {10.2140/agt.2015.15.2861},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.2861/}
}
TY  - JOUR
AU  - Brandenbursky, Michael
AU  - Kędra, Jarek
TI  - Concordance group and stable commutator length in braid groups
JO  - Algebraic and Geometric Topology
PY  - 2015
SP  - 2859
EP  - 2884
VL  - 15
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.2861/
DO  - 10.2140/agt.2015.15.2861
ID  - 10_2140_agt_2015_15_2861
ER  - 
%0 Journal Article
%A Brandenbursky, Michael
%A Kędra, Jarek
%T Concordance group and stable commutator length in braid groups
%J Algebraic and Geometric Topology
%D 2015
%P 2859-2884
%V 15
%N 5
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2015.15.2861/
%R 10.2140/agt.2015.15.2861
%F 10_2140_agt_2015_15_2861
Brandenbursky, Michael; Kędra, Jarek. Concordance group and stable commutator length in braid groups. Algebraic and Geometric Topology, Tome 15 (2015) no. 5, pp. 2859-2884. doi: 10.2140/agt.2015.15.2861

[1] M Brandenbursky, On quasi-morphisms from knot and braid invariants, J. Knot Theory Ramifications 20 (2011) 1397

[2] M Brandenbursky, J Kędra, Ś Gal, M Marcinkowski, Cancelation norm and the geometry of bi-invariant word metrics, Glasgow Math. J. (2015)

[3] D Burago, S Ivanov, L Polterovich, Conjugation-invariant norms on groups of geometric origin, from: "Groups of diffeomorphisms" (editors R Penner, D Kotschick, T Tsuboi, N Kawazumi, T Kitano, Y Mitsumatsu), Adv. Stud. Pure Math. 52, Math. Soc. Japan (2008) 221

[4] D Calegari, scl, MSJ Memoirs 20, Math. Soc. Japan (2009)

[5] J M Gambaudo, É Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems 24 (2004) 1591

[6] E A Gorin, V J Lin, Algebraic equations with continuous coefficients, and certain questions of the algebraic theory of braids, Mat. Sb. 78 (1969) 579

[7] M Kawasaki, Relative quasimorphisms and stably unbounded norms on the group of symplectomorphisms of the eucledean spaces

[8] D Kotschick, Stable length in stable groups, from: "Groups of diffeomorphisms" (editors R Penner, D Kotschick, T Tsuboi, N Kawazumi, T Kitano, Y Mitsumatsu), Adv. Stud. Pure Math. Tokyo, Math. Soc. Japan (2008) 401

[9] C Livingston, A survey of classical knot concordance, from: "Handbook of knot theory" (editors W Menasco, M Thistlethwaite), Elsevier (2005) 319

[10] C Livingston, The stable $4$–genus of knots, Algebr. Geom. Topol. 10 (2010) 2191

[11] D D Long, Strongly plus-amphicheiral knots are algebraically slice, Math. Proc. Cambridge Philos. Soc. 95 (1984) 309

[12] H R Morton, Closed braids which are not prime knots, Math. Proc. Cambridge Philos. Soc. 86 (1979) 421

[13] K Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965) 387

[14] A G Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969) 251

Cité par Sources :