no. 2
Geometric subgroups of surface braid groups
Annales de l'Institut Fourier, Tome 49 (1999) no. 2, pp. 417-472.

Soient M une surface, N une sous-surface et n≤m deux entiers positifs. L’inclusion de N dans M induit un homomorphisme du groupe B n N des tresses à n brins de N dans le groupe B m M des tresses à m brins de M. Nous donnons dans un premier temps des conditions nécessaires et suffisantes pour que cet homomorphisme soit injectif et caractérisons le commensurateur, le normalisateur et le centralisateur de π 1 N dans π 1 M. Ensuite, nous déterminons le commensurateur, le normalisateur et le centralisateur de B n N dans B m M dans les cas où N est un disque et où N est large.

Let M be a surface, let N be a subsurface, and let n≤m be two positive integers. The inclusion of N in M gives rise to a homomorphism from the braid group B n N with n strings on N to the braid group B m M with m strings on M. We first determine necessary and sufficient conditions that this homomorphism is injective, and we characterize the commensurator, the normalizer and the centralizer of π 1 N in π 1 M. Then we calculate the commensurator, the normalizer and the centralizer of B n N in B m M for large surface braid groups.

@article{AIF_1999__49_2_417_0,
     author = {Paris, Luis and Rolfsen, Dale},
     title = {Geometric subgroups of surface braid groups},
     journal = {Annales de l'Institut Fourier},
     pages = {417--472},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
     number = {2},
     year = {1999},
     doi = {10.5802/aif.1680},
     zbl = {0962.20028},
     mrnumber = {2000f:20059},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1680/}
}
TY  - JOUR
AU  - Paris, Luis
AU  - Rolfsen, Dale
TI  - Geometric subgroups of surface braid groups
JO  - Annales de l'Institut Fourier
PY  - 1999
SP  - 417
EP  - 472
VL  - 49
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1680/
DO  - 10.5802/aif.1680
LA  - en
ID  - AIF_1999__49_2_417_0
ER  - 
%0 Journal Article
%A Paris, Luis
%A Rolfsen, Dale
%T Geometric subgroups of surface braid groups
%J Annales de l'Institut Fourier
%D 1999
%P 417-472
%V 49
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1680/
%R 10.5802/aif.1680
%G en
%F AIF_1999__49_2_417_0
Paris, Luis; Rolfsen, Dale. Geometric subgroups of surface braid groups. Annales de l'Institut Fourier, Tome 49 (1999) no. 2, pp. 417-472. doi : 10.5802/aif.1680. https://aif.centre-mersenne.org/articles/10.5802/aif.1680/

[Ar1] E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Hamburg, 4 (1926), 47-72. | JFM

[Ar2] E. Artin, Theory of braids, Annals of Math., 48 (1946), 101-126. | MR | Zbl

[Bi1] J.S. Birman, On braid groups, Comm. Pure Appl. Math., 22 (1969), 41-72. | MR | Zbl

[Bi2] J.S. Birman, Braids, links, and mapping class groups, Annals of Math. Studies 82, Princeton University Press, 1973. | Zbl

[Bi3] J.S. Birman, Mapping class groups of surfaces, Contemporary Mathematics, 78 (1988), 13-43. | MR | Zbl

[Br] K.S. Brown, Cohomology of groups, Springer-Verlag, New York, 1982. | MR | Zbl

[Ch] W. Chow, On the algebraic braid group, Annals of Math., 49 (1948), 654-658. | MR | Zbl

[Co] R. Cohen, Artin's braid groups, classical homotopy theory, and sundry other curiosities, Contemp. Math., 78 (1988), 167-206. | Zbl

[Ep] D.B.A. Epstein, Curves on 2-manifolds and isotopies, Acta Math., 115 (1966), 83-107. | MR | Zbl

[FaN] E. Fadell, L. Neuwirth, Configuration spaces, Math. Scand., 10 (1962), 111-118. | MR | Zbl

[FoN] R.H. Fox, L. Neuwirth, The braid groups, Math. Scand., 10 (1962), 119-126. | MR | Zbl

[FaV] E. Fadell, J. Van Buskirk, The braid groups of E2 and S2, Duke Math. J., 29 (1962), 243-258. | MR | Zbl

[FRZ] R. Fenn, D. Rolfsen, J. Zhu, Centralisers in the braid group and singular braid monoid, L'Enseignement Math., 42 (1996), 75-96. | MR | Zbl

[Ga] F.A. Garside, The braid groups and other groups, Oxford Quart. J. Math., 20 (1969), 235-254. | MR | Zbl

[Go] C.H. Goldberg, An exact sequence of braid groups, Math. Scand., 33 (1973), 69-82. | MR | Zbl

[GV] R. Gillette, J. Van Buskirk, The word problem and its consequences for the braid groups and mapping class groups of the 2-sphere, Trans. Amer. Math. Soc., 131 (1968), 277-296. | MR | Zbl

[LS] R.C. Lyndon, P.E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin, 1977. | MR | Zbl

[Ro] D. Rolfsen, Braid subgroup normalisers and commensurators and induced representations, Invent. Math., 130 (1997), 575-587. | MR | Zbl

[Sc] G.P. Scott, Braid groups and the group of homeomorphisms of a surface, Proc. Camb. Phil. Soc., 68 (1970), 605-617. | MR | Zbl

[Se] J.-P. Serre, Arbres, amalgames, SL2, Astérisque, Soc. Math. France, 46 (1977). | MR | Zbl

[Va] J. Van Buskirk, Braid groups of compact 2-manifolds with elements of finite order, Trans. Amer. Math. Soc, 122 (1966), 81-97. | MR | Zbl

Cité par Sources :