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I believe that there might be an article by Jacques Tits somewhere in which he shows that a locally finite tree can be recovered from the topological group structure on its automorphism group (with the compact-open topology). Because the vertices can be identified with maximal compact subgroups, and one can give a criterion for when two vertices are adjacent.

If anyone happens to know the reference for this I'd be very grateful.

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    $\begingroup$ I could be wrong but, in Section 3 of this paper - perso.uclouvain.be/pierre-emmanuel.caprace/papers_pdf/… - `Tits' independence property' is discussed. This might be what you are looking for. $\endgroup$
    – Nick Gill
    Commented Sep 20, 2012 at 9:03
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    $\begingroup$ You certainly need some regularity assumption, because the automorphism group can be trivial. Probably you want regular or biregular. $\endgroup$
    – YCor
    Commented Sep 20, 2012 at 9:26
  • $\begingroup$ One kind of regularity to impose, not as strong as biregularity, might be cocompactess. But there are counter-examples in that situation too. There exist locally finite trees where the automorphism group acts cocompactly, freely, and properly discontinuously, so all vertex stabilizers are trivial, but the automorphism group is free of finite rank. $\endgroup$
    – Lee Mosher
    Commented Sep 20, 2012 at 17:22

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This is true if the action of the automorphism group is edge-transitive (so in particular the tree has to be biregular). This is precisely the statement of Lemma 2.6(vii and viii) of our paper "Simple locally compact groups acting on trees and their germs of automorphisms", P.-E. Caprace and T. De Medts, Transform. Groups 16, Issue 2 (2011), 375-411, which is also available on arXiv.

We don't claim to be the first ones observing the truth of the statement, but I'm not aware of such a paper by J. Tits either. Perhaps you are thinking about his famous paper "Sur le groupe des automorphismes d’un arbre", Essays on topology and related topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 188–211, but that paper only deals with the abstract group structure, not with the topological group structure.

Added: I should point out that by "the automorphism group" I mean "a (closed) group of automorphisms of the tree", not necessarily the full automorphism group.

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