Locally compact groups acting on discrete structures - Summer School  06 - 10 July 2020

08:00 - 17:00
Room : GA 3 21

Organisers

Alejandra Garrido, The University of Newcastle, Australia

Colin Reid, The University of Newcastle, Australia

Stephan Tornier, The University of Newcastle, Australia

George Willis, The University of Newcastle, Australia

Following recent developments and in accordance with the restrictions imposed by our authorities and the EPFL, this event is cancelled.

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:
  1. Commensurating actions by Yves Cornulier.
  2. 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.

  3. Automorphism groups of trees and graphs by Röggi Möller.
  4. 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.

    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.

  5. Local structure of totally disconnected locally compact groups by Colin Reid.
  6. 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.

  7. Self-similar groups by Zoran Šunić.
  8. 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.

    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.

  9. Endomorphisms and automorphisms of totally disconnected locally compact groups by George Willis.
  10. 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.

Graduate students (as well as postdocs and others interested) are welcome to register by filling in the above registration.

Name University Dates of visit
Raphael Appenzeller ETHZ 06/07/2020 - 10/07/2020
Shivam Arora Memorial University of Newfoundland 06/07/2020 - 10/07/2020
Lara Beßmann University of Münster 06/07/2020 - 10/07/2020
Henry Bradford University of Cambridge 06/07/2020 - 10/07/2020
Alex Carrazedo Dantas Universidade de Brasília 06/07/2020 - 10/07/2020
Sean Carroll The University of Auckland 06/07/2020 - 10/07/2020
Yves Cornulier CNRS, Université Lyon 1 06/07/2020 - 10/07/2020
Thibaut Dumont HEP Vaud 06/07/2020 - 10/07/2020
Amandine Escalier Université Paris Diderot 06/07/2020 - 10/07/2020
Gabriel Favre Stockholm University 06/07/2020 - 10/07/2020
Alejandra Garrido The University of Newcastle, Australia 06/07/2020 - 10/07/2020
Max Gheorghiu University of British Colombia 06/07/2020 - 10/07/2020
Daniel Keppeler Universität Münster 06/07/2020 - 10/07/2020
Elyasheev Leibtag Weizmann Institute 06/07/2020 - 10/07/2020
Christian Lindorfer Graz University of Technology 06/07/2020 - 10/07/2020
Röggi Möller University of Iceland 06/07/2020 - 10/07/2020
Colin Reid The University of Newcastle, Australia 06/07/2020 - 10/07/2020
Tulio Santos Universidade de Brasília 06/07/2020 - 10/07/2020
Stephan Tornier The University of Newcastle, Australia 06/07/2020 - 10/07/2020
Olga Varghese University of Münster 06/07/2020 - 10/07/2020
George Willis The University of Newcastle, Australia 06/07/2020 - 10/07/2020
Zoran Šunić Hofstra University 06/07/2020 - 10/07/2020
Total Guests : 22
Name University Dates of visit
Yves Cornulier CNRS, Université Lyon 1 06/07/2020 - 10/07/2020
Röggi Möller University of Iceland 06/07/2020 - 10/07/2020
Colin Reid The University of Newcastle, Australia 06/07/2020 - 10/07/2020
George Willis The University of Newcastle, Australia 06/07/2020 - 10/07/2020
Zoran Šunić Hofstra University 06/07/2020 - 10/07/2020
Total Guests : 5
Conference in Honor of the 70th Birthday of Tudor Ratiu, 20 to 24 July 2020.
Registration fees of CHF 50.- are mandatory for visits from 1 day to full attendance of the conference.
Cancellations and no-shows are not eligible for a refund.
I hereby agree to the conditions of the registration fees and to settling the amount by 28 February 2020.

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