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This is the first time I am using Mathoverflow and I am still learning how to use it. That is why I want to begin with a curious question:

Does the group of automorphisms of a Bruhat-Tits tree have nontrivial central extensions?

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@Hadi: it seems like a perfectly good qustion, but people might be more interested if you added some motivation. (Also, why is it tagged "ergodic-theory"?) –  Pete L. Clark Oct 12 '10 at 1:15
    
Are you just asking about the full automorphism group of a $q+1$-regular tree (which is quite large), or are you asking for the automorphisms to preserve some algebraic structure? –  S. Carnahan Oct 12 '10 at 3:44
    
@Pete: The automorphism group of a tree and $\mathrm{SL}_2(\mathbb Q_p)$ have similar properties. It is probably natural to guess that their central extensions look similar as well (though it may merely be a wild guess.) There are also a result by Kapoudjian what says that a combinatorial analog of $Diff(S^1)$ has nontrivial central extensions. @Scott: I mean the automorphism group which acts rigidly on edges, hence it is locally compact. –  Hadi Oct 14 '10 at 3:00
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