Scale groups are closed subgroups of the group of isometries of a regular tree that fix an end of the tree and are vertex-transitive. They play an important role in the study of locally compact totally disconnected groups as was recently observed by P-E.Caprace and G.Willis. In the 80th they were studied by A.Figa-Talamanca and C.Nebbia in the context of abstract harmonic analysis and amenability. It is a miracle that they are closely related to fractal groups, a special subclass of self-similar groups. In my talk I will discuss two ways of building scale groups. One is based on the use of scale-invariant groups studied by V.Nekrashevych and G.Pete, and a second is based on the use of liftable fractal groups. The examples based on both approaches will be demonstrated using such groups as Basilica, Hanoi Tower Group and Group of Intermediate Growth (between polynomial and exponential). Additionally, the group of isometries of the ring of integer p-adics and group of dilations of the field of p-adics will be mentioned in the relation with the discussed topics.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006