A normal subgroup theorem for commensurators of lattices

We establish a general normal subgroup theorem for commensurators of lattices in locally compact groups. While the statement is completely elementary, its proof, which rests on the original strategy of Margulis in the case of higher rank lattices, relies heavily on analytic tools pertaining to amenability and Kazhdan's property (T). It is a counterpart to the normal subgroup theorem for irreducible lattices of Bader and the second named author, and may also be used to sharpen that result when one of the ambient factors is totally disconnected.