# Summer term 2022: Discrete Groups, Expanding Graphs and Invariant Measures

Chaired by I. Halupczok, H. Kammeyer, B. Klopsch and M. Zibrowius.

Organised by I. Halupczok

All talks take place on Fridays at 12:30 pm in 25.22.03.73.

If you want to participate but did not get any mails related to this yet, send me a mail so that I put you onto the mailing list.

## Aims and Content - a short description

(partly copy-pasted from Lubotzky's book)
In the last fifteen years, two seemingly unrelated problems were solved by amazingly similar techniques from representation theory and from analytic number theory. One problem is the explicit construction of graphs which are sparse but highly connected. Such graphs are called "expanders"; their existence is easy to prove, but finding explicit constructions is difficult. The other problem is the question whether the Lebesgue measure is, up to scaling, the only rotation invariant, finitely additive measure on the n-sphere.
It turned out that Kazhdan's property (T) (from representation theory of semi-simple Lie groups) was useful to obtain partial results for both problems; later, both problems were solved using a (proved) conjecture by Ramanujan related to automorphic forms.
In the seminar, we will follow the book by Lubotzky called "Discrete Groups, Expanding Graphs and Invariant Measures", which presents the two problems and their solutions from a unified point of view.

## Schedule

(Volunteers still needed for the red talks)
 8.4.22 1. Overview and expanding graphs (Giada) First part [L, Chapter 0]: Give a quick overview over the seminar, following Ch. 0. (The seminar will go up to Chapter 7.) Second part [L, Chapter 1]: Define expanders, families of expanders, and the Cheeger Constant. Prove the existence of expanders (Prop. 1.2.1). Present some of the applications/motivation for expanders from the 2nd half of Sect. 1.1. (Choose whatever sounds interesting to you; choose how much details to give.) 22.4.22 2. The Hausdorff-Banach-Tarski paradox (Martina) [L, Sections 2.0, 2.1]: Define paradoxical group actions; prove that $\operatorname{SO}(3)$ contains a free non-abelian group (Prop. 2.1.7), and use this as an excuse to introduce the quaternions. Deduce Banach-Tarski (Cor. 2.1.14); also present the variant from Thm 2.1.17. 29.4.22 3. Invariant measures (Moritz) [L, Section 2.2]: Introduce amenability, explain Ruziewicz's problem and give first partial results (Props. 2.2.11, 2.2.12). 6.5.22 4. Kazhdan's Property (T) (Margherita) [L, Sections 3.0, 3.1]: Present the motivation from Section 3.0; introduce the Fell topoolgy and Property (T); express amenability in these terms (Thm 3.1.5); present the examples, maybe without proof. 13.5.22 5. Lattices and constructing expanders (Benjamin) [L, Sections 3.2, 3.3]: Introduce the lattices $G(\mathbb Z_S)$ and present whether they have Property (T); apply this to construct expanders 20.5.22 6. Solution to Ruziewicz's Problem for $S^n, n \ge 4$ (Iker) [L, Section 3.4] Find a finitely generated dense subgroup of $\operatorname{SO}(n)$ with Property (T) (Prop 3.4.3) and deduce the title of the talk (Thm 3.4.2); present some of the generalizations and variants following later in the section (maybe without proofs) 27.5.22 7. Laplacians (Luca) [L, Sections 4.0-4.2:] Introduce Laplacians of manifolds (Sect. 4.1) and combinatorial Laplacians of graphs (Sect. 4.2). Relate to the Cheeger constant from the first talk. 3.6.22 8. Selberg's Theorem and Ramanujan graphs (Holger) First part [L, Sections 4.3, 4.4]: Explain how Selberg's Theorem (Thm 4.4.1) can be used to obtain more expanders. Only do as much as necessary for that from Sect. 4.3; in particular, you might have to skip most of the proof of Thm 4.3.2 and maybe even most of its statement. Seond part [L, Section 4.5]: Introduce random walks on graphs, relate this to being an expander and introduce the notion of Ramanujan graph (Don't spend too much time on technicalities, but try to convey why Ramanujan graphs are interesting.) 10.6.22 9. Representation theory of $\operatorname{PSL}_2(\mathbb R), \operatorname{GL}_2(\mathbb R), \operatorname{PGL}_2(\mathbb Q_p), \operatorname{GL}_2(\mathbb Q_p)$ (Florian) [L, Sections 5.0-5.2]: Give an introduction to the representation theory of those groups 17.6.22 10. $\operatorname{PGL}_2(\mathbb Q_p)$ and the Selberg Conjecture (tba) First part [L, Sections 5.3,5.4]: Introduce the trees and the Hecke-operators associated to $\operatorname{PGL}_2(\mathbb Q_p)$ Second part [L, Section 5.5]: Present Theorem 5.5.1 and show how it implies the Selberg Conjecture (Corollary 5.5.2) 24.6.22 11. $L^2(G(\mathbb Q)\setminus G(\mathbb A))$ (Hamed) [L, Chapter 6]: Introduce Adèles, quaternion algebras and the strong approximation theorem; this talk is more a kind of survey, just explaining results and not giving proofs. 1.7.22 12. The Banach-Ruziewicz problem for $n = 2,3$ (tba) [L, Sections 7.0-7.2]: Prove the Banach-Ruziewicz problem for $n = 2,3$ (Thm 7.2.1) 8.7.22 13. Ramanujan Graphs (tba) [L, Sections 7.3,7.4]: Construct Ramanujan Graphs (Thm 7.3.1; Sect 7.3). If time permits, explain how the construction can be made more explicit (Sect. 7.4). 15.7.22 14. Discussion for next semester's seminar

## Literature

The main reference is:
[L] A. Lubodzky: Discrete Groups, Expanding Graphs and Invariant Measures Some more stuff about expander graphs can be found in:
[K] E. Kowalski: An introduction to expander graphs

## Archive

WS 2021/22: Superrigidity

SS 2021: Group cohomology

SS 2020 and WS 20/21: cancelled due to pandemic

WS 2019/20: Intersection theory

SS 2019: Knots and primes

WS 2017/18: Algebraic K-theory

SS 2017: Berkovich spaces

SS 2016: Modular Representation Theory

WS 15/16: The Milnor Conjectures

SS 2015: Rationality

WS 14/15: Essential Dimension

SS 2014: Varieties of Representations

Website edited by Holger Kammeyer (2021)