# Finite outer automorphism groups

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?

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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.

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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 front.math.ucdavis.edu/0601.5305 –  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 front.math.ucdavis.edu/0605.5553. 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.

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@Richard: thanks! –  Igor Rivin Apr 25 '11 at 20:17