On the Automorphism Group of the One-Rooted Binary Tree

AbstractLetAbe the automorphism group of the one-rooted regular binary treeT2andGthe subgroup ofAconsisting of those automorphisms admitting a “finite description” in their action onT2. LetNA(G) be the normaliser ofGinA, let Aut(G) be the group of automorphisms ofG, and let EndA(G) be the semigroup of endomorphisms ofGinduced by conjugation by elements ofA. ThenGis the infinite iterated wreath product (…≀C2)≀C2, andAis the topological limit ofG. We study in some detail the structure ofG, Aut(G), and EndA(G). In particular, we proveNA(G) is isomorphic to Aut(G), contains a copy ofAitself, and is a proper subgroup of EndA(G). Furthermore we discuss connections with automata and introduce the notion of functionally recursive automorphisms ofT2