MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

It is a theorem of Gopal Prasad (which I hope I am not misquoting...) that lattices in higher rank linear semi-simple Lie groups have finite outer automorphism groups. Is there some other reasonable class of groups with this finiteness property?

share|cite|improve this question

Here is a result of Fr\'ed\'eric Paulin: a hyperbolic group with Kazhdan's property (T), has a finite outer automorphism group. See Outer automorphisms of hyperbolic groups and small actions on R-trees. Arboreal group theory (Berkeley, CA, 1988), 331–343, Math. Sci. Res. Inst. Publ., 19, Springer, New York, 1991.

share|cite|improve this answer
Thanks! Of course this begs the question of where such groups would come from... – Igor Rivin Apr 25 '11 at 19:24
More generally, relatively hyperbolic groups that don't split over elementary subgroups have this property by a result of Drutu-Sapir, see Theorem 1.12 in – Igor Belegradek Apr 25 '11 at 19:28
@Igor Rivin: there are lots of relatively hyperbolic groups with no elementary splittings. What kind of groups are you looking for? – Igor Belegradek Apr 25 '11 at 19:30
@Igor Rivin: any countable group embeds into some Out(G) where G has property (T), see In fact, I seem to recall that Minasyan-Osin showed that any countable group can be realized as Out(G) where G has property (T), but I cannot find a reference at the moment. – Igor Belegradek Apr 25 '11 at 19:41
@Igor Rivin: I meant to say that a relatively hyperbolic group G that doesn't split over elementary subgroups must ahve finite Out(G). – Igor Belegradek Apr 25 '11 at 19:42

The theorem Igor Belegradek mentions is a more general form of the theorem (due collectively to Bestvina, Feighn, Paulin, and Rips) that a one-ended hyperbolic group has infinite outer automorphism group only if it splits over a two-ended subgroup. See

M. Bestvina, M. Feighn, Stable actions of groups on real trees. Invent. Math. 121 (1995), no. 2, 287-321.

share|cite|improve this answer
@Richard: thanks! – Igor Rivin Apr 25 '11 at 20:17

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.