Abstract commensurators of braid groups, with D. Margalit
 

abstract
 

Let B_n be the braid group on n > 3 strands and Mod(S) 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. We also show that Tv is itself isomorphic to a semidirect product of an infinite dimensional rational vector space with the multiplicative group of non-zero rational numbers.