On the lattice of normal subgroups in ultraproducts of compact simple   groups
  We prove that the lattice of normal subgroups of ultraproducts of compact simple non-abelian groups is distributive. In the case of ultraproducts of finite simple groups or compact connected simple Lie groups of bounded rank the set of normal subgroups is shown to be linearly ordered by inclusion. 
Abel Stolz and Andreas Thom
math.GR math.LO