This is a website for the reading group on Stability that we are doing in the Winter semester 2022/23 in Ljubljana.
Each participant chooses a topic and prepares a one-hour lecture for the group. After the topics are chosen, we will discuss possible slots in the Winter semester with the organizers of the SAFA seminar. You do not need to cover the whole material in great detail and it is completely OK to only explain the main ideas (after all, you only have 1 hour). For most of these topics, you can also find the corresponding video lecture at this link.
If you decide to participate, send an email to urban.jezernik@fmf.uni-lj.si so that I mark the topic has been chosen.
(Primož Moravec) Stability of finite groups: Explain why finite groups are stable in permutations [Glebsky-Rivera 2009]. [paper]
(Primož Potočnik) Stability of Z^2 in permutations: Explain the proof of the theorem [Arzhantseva-Paunescu 2015] on stability of the equation AB=BA in permutations with the normalized Hamming distance. [paper]
(Lucijan Plevnik) Stability of Z^2 in unitary groups wrt rank distance: Explain the proof of [Elek-Grabowski 2021] on stability of the equation AB=BA in unitary groups with the normalized rank distance. [paper]
(Nikola Kovačević) Sofic groups: Introduce sofic groups, show some equivalent definitions, show the main ideas in the proof of [Elek-Szabo 2004] that Kaplansky's direct finiteness conjecture holds for sofic groups. [msc thesis, paper]
(Urban Jezernik) Non-approximable groups: Present the main ideas and tools in the proof of existence of non-Frobenius approximable groups by [de Chiffre, Glebsky, Lubotzky, Thom 2020]. [paper]
Testability: Introduce testability, show its connections with stability, introduce amenable groups, explain why amenable groups are testable. [paper]
IRS: Introduce Invariant Random Subgroups, explain how they are related to stability for amenable groups. [slides, paper]
Stability and property (T): Show why sofic groups with property (T) are not stable, and possibly expand by including that they are not even testable. [paper1, paper2]
Flexible stability: Introduce flexible stability, show how proving flexible stability for SLn(Z) for some n would give an example of a non-sofic group. [paper]
(Aleš Vavpetič) Surface groups: Show that surface groups are flexibly stable. [paper]
(Ganna Kudryavtseva) Grigorchuk group: Introduce the Grigorchuk group and show that it is stable. [paper]
We meet Tuesdays at 13.15 in classroom 2.02.
13/10/2022 Urban Jezernik: Stability & Testability [notes]
08/11/2022 Primož Moravec: Stability of finite groups [notes]
15/11/2022 Nikola Kovačević: Sofic groups [notes]
22/11/2022 Aleš Vavpetič: Surface groups, part 1
29/11/2022 Aleš Vavpetič: Surface groups, part 2 [notes]
06/12/2022 Ganna Kudryavtseva: Grigorchuk group [notes]
13/12/2022 Urban Jezernik: Non-approximable groups [notes]