Seminar: Small cancellation and applications

Seminar angeboten von: Prof. Dr. Dr. Katrin Tent, Jun.-Prof. Dr. Tim de Laat, Prof. Dr. Siegfried Echterhoff und Dr. Aleksandra Kwiatkowska.

Zeit/Ort:

Das Seminar findet donnerstags 14-16 Uhr in SR1D statt.

Beschreibung:

The theory of groups with small cancellation provides powerful methods to construct groups with exotic properties. Recently, such methods have been used successfully in order to establish the existence of Gromov monster groups, and more specifically, in order to construct several classes of such monster groups with prescribed properties. The existence of such groups has far-reaching consequences in the area of operator algebras and noncommutative geometry. For example, such groups have lead to counterexamples to the Baum-Connes conjecture which computes the K-theory of crossed products of C*-dynamical systems.

The main focus of this seminar is the article of Osajda [Os] (see also [AO]), in which Osajda gives explicit constructions of Gromov monster groups. One important feature of these groups is that they isometrically contain expanders in their Cayley graphs. Expanders are increasing and unbounded sequences of highly connected sparse graphs, which have lead to striking consequences in mathematics and theoretical computer science.

After an introduction to the theory of small cancellation and graphical small cancellation, we will discuss expanders and constructions of expanders from groups with property (T). After that, we will discuss the article [Os] in detail. At the end, we will discuss how the groups constructed by Osajda lead to counterexamples to the Baum-Connes conjecture.

Literatur:

- [AO] G. Arzhantseva and D. Osajda, Infinitely presented small cancellation groups have the Haagerup property.

J. Topol. Anal. 7 (2015), 389-406.

- [DK] C. Drutu and M. Kapovich, Geometric Group Theory. American Mathematical Society, Providence, RI, 2018.

- [G] G. Gardam, Expander Graphs and Kazhdan’s Property (T), B. Sc. Thesis, University of Sydney, 2012.

- [LS] R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory. Reprint of the 1977 edition, Springer-Verlag, Berlin, 2001.

- [O] Y. Ollivier, On a small cancellation theorem of Gromov. The Bulletin of the Belgian Mathematical Society - Simon Stevin, 13(1) Nov. 2003

- [Os2] D. Osajda, Residually finite non-exact groups. GAFA, April 2018, Volume 28, Issue 2, pp 509-517.

- [Os1] D. Osajda, Small cancellation labellings of some infinite graphs and applications. Preprint (2014), arXiv:1406.5015.

- [RS] E. Rips and Y. Segev, Torsion-free group without unique product property. J. Algebra 108 (1987), 116-126.

Vortragsliste:

1. Small cancellation groups [LS, Chapter V] (Vortrag: Antonia Klein Reesink)

2. Hyperbolicity [pdf, Seite 29](Vortrag: Aleksandra Kwiatkowska)

3. Graph cancellation [LS, Chapter V, Section 9] (Vortrag: Stefan Ludwig)

4. Property (T) [G](Vortrag: Julian Kranz)

5. Expander graphs (Vortrag: Felix Hildebrandt)

6-8. Osajda's article [Os1] (Vorträge: Ina Vogler, Leon Pernak (notes), Markus Schmetkamp)

9. Osajda's article [Os2] (Vortrag: Timo Siebenand)

10. Baum-Connes conjecture (Vortrag: Manuel König)