turkmath.org

One may think that the group of automorphisms of a tree, being a profinite group, is something which resembles the absolute Galois group and as such is not good for your mental health. Here what we mean by a tree is a regular tree, that is a tree with fixed vertex degrees (=d). Denote this tree by A_d. In case d is at least 3, the automorphism group is uncountable. My aim is to describe each element of this group as a "shuffle". Then I will give another description as a "twist". It is very easy to see each element explicitly in these descriptions, but the group multiplication is somewhat messy. To this end, I will have to endow the tree with the structure of a plane tree (nevertheless the automorphisms will be tree automorphisms, not plane tree automorphisms) After this I will define the boundary of the tree and look at the action of the group on this boundary. Plane tree structure induces a cyclic order structure on the boundary of the tree. If we glue those points which are not separated by a third point, then we obtain a space which is homeomorphic to the circle. The automorphism group does not respect the order structure but nevertheless acts on the circle "in some way". My ultimate aim is to understand this action.