Automorphisms of rooted trees.

Abstract

Groups acting on p-adic trees have been well studied over the past decades since they represent a source of examples with interesting properties in group theory. Groups with intermediate growth or counterexamples to the General Burnside Problem can be found inside this class of groups. In this thesis we analyze some properties concerning the structures of two families of groups acting over primary regular rooted trees, i.e. regular rooted trees such that every vertex has a number of descendants equal to a power of a prime. These two families are the GGS-groups acting over the p^n-adic tree and the p-Basilica groups, a generalization of the Basilica group over the p-adic tree for a prime p. A GGS-group over the p^n-adic tree is defined by a vector e whose components are elements in Z/p^nZ.Depending on the defining vector of the GGS-group, we determine which of them are branch. We reduce our study to the fractal GGS-groups, since the non-fractal ones cannot be branch. We prove that all of them, except the ones acting over the 2^n-adic tree whose defining vectors have only one invertible component in position 2^{n-1}, are weakly regular branch. The GGS-groups with constant defining vector are weakly regular branch but not branch. For the other GGS-groups, we prove that they are all regular branch with some small exceptions for which the question is still open. The p-Basilica groups are weakly branch but not branch for any prime p. These provide the first examples of groups with these properties that are super strongly fractal. For this class of groups we also study other problems. We show that they have the p-congruence subgroup property but not the congruence subgroup property nor the weak congruence subgroup property, providing the first examples of weakly branch groups with such properties. We compute the orders of the congruence quotients of these groups, which enables us to determine the Hausdorff dimensions of the p-Basilica groups. Lastly, we show that the p-Basilica groups do not possess maximal subgroups of infinite index and that they have infinitely many non-normal maximal subgroups.