My friend asked me to ask his question here. Where he can find (a paper or a book) containing a complete description (with the proof) of a structure of the group of automorphisms of an infinite binary tree?



I am assuming you mean a binary rooted tree. I don't know your definition of complete. The group is the infinite permutational wreath product of symmetric groups of degree 2. Good references are Automata, dynamical systems and infinite groups by R. Grigorchuk, V.V.Nekrashevich, V.I.Sushchanskii, Proc. Steklov Inst. Math. v.231 (2000), 134-214 and Cyclic renormalization and automorphism groups of rooted trees by Bass, Otero-Espinar, and Rockmore.

  • $\begingroup$ One more thing: The (almost) full automorphism group $Aut_+(T)$ is an HNN extension of $Aut(T,r)$ (the group described in your answer), where $r$ is a fixed vertex of $T$ (a "root"). Here $Aut_+(T)$ is the subgroup of $Aut(T)$ of index 2 consisting of automorphisms acting without inversions. The edge subgroup here is also well-understood, it is isomorphic to the direct product $Aut(T,r)\times Aut(T,r)$. $\endgroup$ – Misha Feb 4 '14 at 8:52
  • $\begingroup$ @Misha, If the OP means by binary tree the Cayley graph of the free monoid on 2-generators, then the root is the only degree 2 vertex and so must be fixed. $\endgroup$ – Benjamin Steinberg Feb 4 '14 at 15:14
  • $\begingroup$ I see, I was thinking about the regular trivalent tree. $\endgroup$ – Misha Feb 4 '14 at 18:58
  • $\begingroup$ The op was not clear on this point. $\endgroup$ – Benjamin Steinberg Feb 4 '14 at 19:13
  • $\begingroup$ @BenjaminSteinberg Thanks a lot. My friend said that you are a well-known specialist in this area and he is happy with your answer. $\endgroup$ – Alex Ravsky Feb 9 '14 at 6:10

Topics in geometric group Theory by P. de la Harpe should also contain this and is very well written.

  • $\begingroup$ This is also a good one. $\endgroup$ – Benjamin Steinberg Feb 3 '14 at 23:32

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