Tree actions associated with the scale of an automorphism of a totally disconnected locally compact group

One of the key features of the structure theory of totally disconnected locally compact groups is the existence of certain compact open subgroups, called tidy subgroups, which are well-behaved under the action of group automorphisms. If V is a compact open subgroup that is tidy for the automorphism α then there is an associated closed subgroup V__ which is invariant under α. These V__ groups are analogous to eigenspaces for linear operators in the theory of Lie groups. There is a representation of the semi-direct product V__ ⋊ 〈α〉 as a closed subgroup of the stabiliser of an end of a homogeneous tree, and it is this tree-representation that we aim to understand in this thesis. First, we use the properties of the tree-representation to reduce the problem to understanding the automorphism groups of rooted trees that have a self-similarity property which we call R. These groups are compact and hence profinite, which means we can understand them in terms of their finite quotients which have a corresponding property which we call R_n. Then we use the software package MAGMA to perform calculations with these finite groups, generating plenty of examples and providing evidence in support of several conjectures about the behaviour of groups with property R. Finally, we describe two general constructions, both of which take a finite group with property Rn and extend it to a profinite group with property R. One construction generates the maximal such group, which turns out to be a type of self-similar group called a finitely constrained group. We show that all groups with property R can be approximated by these finitely constrained groups. The other construction uses finite automata to produce topologically finitely generated groups with property R.