Abstract commensurators of braid groups
Christopher J. Leininger, Dan Margalit
Abstract
Let Bn 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 Bn is isomorphic to a semidirect product of Mod(S) with a group we refer to as the transvection subgroup, Tv(Bn). We also show that Tv(Bn) 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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