This Summer School is intended as an introduction to the main themes of the program, focusing on key topics of the subsequent workshops: automorphism groups of graphs, dynamics on totally disconnected locally compact groups and their relation to self-similar groups, local structure of totally disconnected locally compact groups, and geometry and dynamics.Speakers include:
- Yves Cornulier (CNRS and Université de Lyon 1)
- Röggi Möller (University of Iceland)
- Colin Reid (University of Newcastle, Australia)
- Zoran Šunić (Hofstra University)
- George Willis (University of Newcastle, Australia)
The school will consist of mini-courses complemented by problem and discussion sessions.Topics include:
- Commensurating actions by Yves Cornulier. I will introduce the notion of commensurating action for a topological group, and the corresponding rigidity property: Property FW. It turns out that a topological group has Property FW if and only if every action of it on a nonempty CAT(0) cube complex has a fixed point. I will describe applications of this notion, as well as the notion of partial action, to the study of groups of piecewise continuous transformations in dimension 1.
- Automorphism groups of trees and graphs by Röggi Möller. The automorphism group of a locally finite connected graph carries a topology, the topology of pointwise convergence, that makes it into a totally disconnected locally compact group. In this topology the stabilizer of a vertex is a compact open subgroup. Conversely, if we start with a totally disconnected locally compact group one can construct a locally finite connected graph that the group acts on such that the stabilizer of a vertex is a compact open subgroup. This graph is commonly called the Cayley-Abels graph of the group.
- Local structure of totally disconnected locally compact groups by Colin Reid. In a topological group, the local structure is the structure that is present in every neighbourhood of the identity. For totally disconnected locally compact groups, the connection between local and global structure is more subtle than it is for Lie groups, but still gives important information. I will be giving an overview of the local structure theory of totally disconnected locally compact groups, including the existence of arbitrarily small profinite open subgroups (Van Dantzig's Theorem); the universal group of a given local isomorphism type; and the consequences of local decomposition properties.
- Self-similar groups by Zoran Šunić. The set of automorphisms of a regular rooted tree carries three structures: group, metric, and self-similarity. A natural approach then, in the study of groups of tree automorphisms, is to simultaneously employ algebraic, topological, and symbolic dynamics methods. The interplay gives rise to deep and beautiful theory and produces important examples contributing to our understanding of groups and their actions from different points of view.
- Endomorphisms and automorphisms of totally disconnected locally compact groups by George Willis. The dynamics of the action of an endomorphism on a t.d.l.c. group may be characterised through invariants such as the scale of the endomorphism, and described by structures in the group such as subgroups tidy for the endomorphism and its contraction subgroup. These invariants serve to analyse the group when calculated for inner automorphisms. I will illustrate these ideas with numerous examples before going on to say more about how they apply when several automorphisms are considered simultaneously. The key concept here is that of flatness of a group of automorphisms, and results about at groups will be developed in more detail.
The Cayley-Abels graph carries information about the structure of the group in much the same ways as the Cayley graph of a finitely generated group. In these talks I will discuss the interplay between the structure of the group and properties of the graph and the action of the group on the graph and touch upon a wide variety of concepts, such as the modular function, the scale function, growth and Stallings' End Theorem. Of particular interest is the case when the graph is a tree and I will discuss various topics in the theory of group actions on trees, such the Burger-Mozes universal groups and Simon Smith's box product.
Natural sources of finitely generated self-similar groups, such as iterated monodromy groups and, more generally, groups generated by finite self-similar sets (often called automaton groups) will be presented. The usual suspects, adding machine, dihedral, lamplighter, Grigorchuk, Basilica, Hanoi, ... along with a few new examples, will be introduced in sufficient depth to allow understanding of the properties that make them so interesting/significant.
We will also discuss topologically finitely generated, compact, self-similar groups that arise as closures of self-similar groups. In particular, finitely constrained groups, that is, groups that are tree-shifts of finite type, will be considered.
The theory of closed self-similar groups fits in the more general framework of the theory of totally disconnected locally compact groups both as an important special case and as a tool that can help in the understanding of the bigger picture.