Lavrenyuk Y. Homeomorphism groups of Cantor spaces

We investigate local isometry groups of Cantor spaces. It is proved that automorphism group of saturated weakly branch group is isomorphic to the normalizer of this group in the isometry group of the corresponding rooted tree. We construct faithful action of the free product of two powers of the infinite cyclic group on the binary rooted tree by recursive automatic automorphisms. We prove that if the group of local isometries of an infinite compact ultrametric space is transitive, then it is complete. Classification of the groups of local isometries of the spherically homogeneous rooted tree boundaries is established. The lattice of normal subgroups is described for such groups. It is proved that these groups are ambivalent and complete. Classification of the groups of measure-preserving self-homeomorphisms of spherically homogeneous tree boundaries is established. The normal structure of LDA-groups is established. We classify LDA-groups which are defined by thick Bratteli diagrams. Simple LDA-groups form important class of non-finitary simple locally finite groups. The full classification of simple LDA-groups is given.