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).

  • $\begingroup$ Central subgroups only (if you take lattices in semi-simple connected Lie groups)? $\endgroup$ – Mark Sapir Apr 18 '11 at 2:46
  • $\begingroup$ @Mark: That's what I thought, but was always puzzled why no one actually stated it this way, so thought that maybe I was missing something... $\endgroup$ – Igor Rivin Apr 18 '11 at 2:49

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

  • 4
    $\begingroup$ The fact that finite normal subgroups $N$ in lattices are central in the ambient Lie group, follows by invoking Zariski density of lattices (to the effect that $N$ is actually normal in the ambient Lie group), and then the classical fact that a discrete normal subgroup in a connected group must be central. $\endgroup$ – Alain Valette May 2 '11 at 8:46

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