# zbMATH — the first resource for mathematics

Abstract commensurators of braid groups. (English) Zbl 1103.20034
Summary: Let $$B_n$$ be the braid group on $$n\geqslant 4$$ strands. We show that the abstract commensurator of $$B_n$$ is isomorphic to $$\text{Mod}(S)\ltimes(\mathbb{Q}^\times\ltimes\mathbb{Q}^\infty)$$, where $$\text{Mod}(S)$$ is the extended mapping class group of the sphere with $$n+1$$ punctures.

##### MSC:
 20F36 Braid groups; Artin groups 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 57M07 Topological methods in group theory
Full Text:
##### References:
 [1] Bell, R.W.; Margalit, D., Injections of Artin groups · Zbl 1148.20024 [2] Birman, J.S., Braids, links, and mapping class groups, Ann. math. stud., vol. 82, (1974), Princeton Univ. Press Princeton, NJ [3] Birman, J.S.; Brendle, T.E., Braids: A survey, in press, in: W. Menasco, M. Thistlethwaite (Eds.), Handbook of Knot Theory · Zbl 1094.57006 [4] Brendle, T.E.; Margalit, D., Commensurations of the Johnson kernel, Geom. topol., 8, 1361-1384, (2004) · Zbl 1079.57017 [5] Charney, R.; Crisp, J., Automorphism groups of some affine and finite type Artin groups, Math. res. lett., 12, 2-3, 321-333, (2005) · Zbl 1077.20055 [6] Crisp, J., Automorphisms and abstract commensurators of 2-dimensional Artin groups, Geom. topol., 9, 1381-1441, (2005) · Zbl 1135.20027 [7] Dyer, J.L.; Grossman, E.K., The automorphism groups of the braid groups, Amer. J. math., 103, 6, 1151-1169, (1981) · Zbl 0476.20026 [8] B. Farb, M. Handel, Commensurations of $$\mathit{Out}(F_n)$$ · Zbl 1137.20018 [9] Farb, B.; Ivanov, N.V., The Torelli geometry and its applications, research announcement, Math. res. lett., 12, 2-3, 293-301, (2005) · Zbl 1073.57012 [10] B. Farb, S. Weinberger, Hidden symmetry and arithmetic manifolds, in: Robert Brooks Memorial Conference proceedings, in press [11] Godelle, E., Normalisateur et groupe d’Artin de type sphérique, J. algebra, 269, 1, 263-274, (2003) · Zbl 1060.20032 [12] Ivanov, N.V., Automorphisms of complexes of curves and of Teichmüller spaces, Int. math. res. not., 1997, 14, 651-666, (1997) · Zbl 0890.57018 [13] Korkmaz, M., Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology appl., 95, 2, 85-111, (1999) · Zbl 0926.57012 [14] Neumann, W.D.; Reid, A.W., Arithmetic of hyperbolic manifolds, (), 273-310 · Zbl 0777.57007 [15] Paris, L., Parabolic subgroups of Artin groups, J. algebra, 196, 2, 369-399, (1997) · Zbl 0926.20022 [16] Rolfsen, D., Braid subgroup normalisers, commensurators and induced representations, Invent. math., 130, 3, 575-587, (1997) · Zbl 0897.20031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.