Totally Disconnected, Locally Compact Groups

The course takes place on Fridays, 3-5 pm, in HG G 26.5.

On May 17 and May 24 the lecture will be until 18:00. There will be no lecture on May 31.

Here we provide an outline of the course content and references where the content of the course can be looked up. Some of it might only be accessible via the ETH network (e.g. while connected to eduroam, or via VPN). Note that the given material never exactly corresponds to what was done in class. If you find references that you find better, I can put them here. If you have any questions or comments, don't hesitate to contact me.

February 22: Motivation and the automorphism group of the tree
W. Lederle's PhD thesis, Section 3.4 until (and without) Section 3.4.1

March 1: Generalities on topological groups, existence of group topologies from a neighbourhood basis of the identity element
Bourbaki: General Topology, Chapter 3
Section 1 TOPOLOGIES ON GROUPS: Subsection 1 Topological groups (without the last paragraph about mappings from X to G), Subsection 2 Neighbourhoods of a point in a topological group
Section 2 SUBGROUPS, QUOTIENTS GROUPS, HOMOMORPHISMS, HOMOGENEOUS SPACES, PRODUCT GROUPS: Subsection 1 Subgroups of a topological group, Proposition 7, Proposition 8, Proposition 16, Proposition 18, Proposition 20, Corollary of Proposition 22, Proposition 23

March 8 : Proof of Van Dantzig's Theorem, profinite groups
New directions in locally compact groups, Chapter 1 (by Marc Burger), Section 1.1

March 15: Profinite groups (continued), examples: infinite Galois groups, vertex stabilizers of tree automorhpism groups
Profinite groups, by Ribes and Zalesskii, Chapter 1: Inverse and Direct Limits, Chapter 2: Profinite groups (until and including Theorem 2.1.3), Subsection 2.11 Profinite Groups as Galois Groups (until and including Theorem 2.11.1)
An introduction to totally disconnected, locally compact groups, by Phillip Wesolek Theorem 1.24

March 22: p-adic groups
p-adic analysis compared to real, Section 1.4 until and including Theorem 1.30, all the necessary definitions before that (without proofs)

March 29: Neretin's group, Higman-Thompson groups
W. Lederle's PhD thesis, Chapter 5 until (and including) Example 5.1.9
Coloured Neretin Groups, by W. Lederle, Section 2.4; Theorem 4.1 with proof (forget about the F in the subscripts)

April 5: Neretin's group: simplicity
Simplicity of Neretin's group of spheromorphisms, Corolarry 4.1 with proof (only in the case of Neretin's group)
Haar Measures, by Stephan Tornier, Definition 2.1, Theorem 2.2 (without proof), Definition 4.7, Proposition 4.8 (without proof), Definition of a lattice (sentence above Proposition 4.12)
Simple groups without lattices, by Bader, Caprace, Gelander and Mozes, until and with Section 3

April 5: Neretin's group: simplicity
Simplicity of Neretin's group of spheromorphisms, Corolarry 4.1 with proof (only in the case of Neretin's group)
Haar Measures, by Stephan Tornier, Definition 2.1, Theorem 2.2 (without proof), Definition 4.7, Proposition 4.8 (without proof), Definition of a lattice (sentence above Proposition 4.12)
Simple groups without lattices, by Bader, Caprace, Gelander and Mozes, until and with Section 3

April 12: Neretin's group: no lattices
W. Lederle's PhD thesis, Lemma 9.1.3.
Simple groups without lattices, by Bader, Caprace, Gelander and Mozes, until and with Section 3

May 3: Cayley-Abels graphs
Notes by Phillip Wesolek, Proposition 4.3, Theorem 4.4, Lemma 4.7

May 3: Cayley-Abels graphs, continued
Notes by Phillip Wesolek, Proposition 4.6, Lemma 4.8, Theorem 4.9

May 10: Local prime content
Locally compact groups built up from p-adic Lie groups, by Helge Glöckner, Definition of the local prime content and first sentence of page 449
Compact open subgroups in simple totally disconnected groups, by George Willis, Lemma 2.3
Locally normal subgroups of totally disconnected groups. Part II, by Caprace, Reid and Willis, Proposition 4.6 (only the part concerning the local prime content, only the special case of topologically simple groups)
Examples: pro-p groups, p-adic groups, automorphism group of a tree, the direct product of (i) all cyclic groups of finite order (ii) all cyclic groups of prime order

May 17: Locally normal subgroups
Locally normal subgroups, local equivalence and the structure lattice, notes by Maxime Gheysens, Section 4.1, Defintion 4.2, Section 4.3, Addendum
Locally normal subgroups of totally disconnected groups. Part II, by Caprace, Reid and Willis
Examples

May 24: The structure lattice, an embedding restriction theorem for locally normal subgroups
Locally normal subgroups, local equivalence and the structure lattice, notes by Maxime Gheysens, Definition 4.5, 4.4.1, 4.4.3, 4.4.2
Groups acting on trees: from local to global structure, by Burger and Mozes, Example 1.1.2
The structure lattice of a t.d.l.c. group, survey by John S. Wilson, definitions of the quasi-center and the quasi-centralizer, Lemma 17.4 ("nilpotent" replaced by "abelian" in the hypothesis and "with non-trivial center" in the conclusion), Lemma 17.5(c), Lemma 17.6, Theorem 17.7 ("soluble" replaced by "abelian")