Abstract commensurators of braid groups

2005/01/26 by Christopher J. Leininger, Dan Margalit, Leininger, Christopher J. +1
Mathematics · #20F28 #20F36 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR #math.GT #msc:20F28 #msc:20F36

paper · pdf · doi:10.48550/arxiv.math/0501480

10 pages

arxiv created 2005/01/26 · openalex publication_date 2005/01/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B_n be the braid group on n strands, with n at least 4, and let Mod(S) be the extended mapping class group of the sphere with n+1 punctures. We show that the abstract commensurator of B_n is isomorphic to a semidirect product of Mod(S) with a group we refer to as the transvection subgroup, Tv(B_n). We also show that Tv(B_n) is itself isomorphic to a semidirect product of an infinite dimensional rational vector space with the multiplicative group of nonzero rational numbers.

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