Margulis' normal subgroup theorem states that any normal subgroup of a higher rank lattice is either finite or of finite index. The obvious question is: can one classify finite normal subgroups of such lattices? (even $SL(n, \mathbb{Z})$ and $Sp(2n, \mathbb{Z})$ would be a good start).

These are the central subgroups, see http://www.mathematik.uniregensburg.de/loeh/seminars/normal_subgroup_thm.pdf . It is proved that every noncentral normal subgroup has finite index (page 7). 

