Normal subgroups of the braid group and the metaconjecture of Ivanov

2018/01/16 by Alan McLeay, McLeay, Alan · 1 citation
Mathematics · #20F28 #20F36 #20F65 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1801.05209

openalex publication_date 2018/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that many normal subgroups of the braid group modulo its centre, and of the mapping class group of a sphere with marked points, have the property that their automorphism and abstract commensurator groups are mapping class groups of such spheres. As one application, we establish the automorphism groups of each term in the lower central series and derived series of the pure braid group. We also obtain new proofs of results of Dyer-Grossman and Orevkov, showing that the automorphism groups of the braid group and of its commutator subgroup are isomorphic. We then calculate the automorphism and abstract commensurator groups of each term in the hyperelliptic Johnson filtration, recovering a result of Childers for the Torelli case. The techniques used in the paper rely on resolving a metaconjecture of Nikolai V. Ivanov for "graphs of regions", extending work of Brendle-Margalit.

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• Geometric normal subgroups in mapping class groups of punctured surfaces

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