Geometric subgroups of surface braid groups

Paris, Luis; Rolfsen, Dale

Annales de l'Institut Fourier, Tome 49 (1999) no. 2, pp. 417-472.

Soient $M$ une surface, $N$ une sous-surface et $n\le m$ 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 ${\pi }_{1}N$ dans ${\pi }_{1}M$. 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 $n\le m$ 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 ${\pi }_{1}N$ in ${\pi }_{1}M$. Then we calculate the commensurator, the normalizer and the centralizer of $B_{n}N$ in $B_{m}M$ for large surface braid groups.

[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). | Numdam | 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