This summer school on high-dimensional expanders is the first event organized in the framework of the EOS-project "High-dimensional expanders and Kac–Moody–Steinberg groups". It consists of 7 short series of lectures by experts in the field and is meant to be accessible to a wide audience of both young and established researchers.

Topics include:

• Ramanujan graphs and expander graphs
• Bruhat–Tits buildings
• Kazhdan's property (T)
• Hecke algebras
• Lattices in p-adic groups
• Ramanujan complexes and high-dimensional expanders
• Applications

Alain Valette: Ramanujan graphs

The mini-course will start with a recap on algebraic graph theory (the adjacency matrix of a graph and the combinatorial significance of its eigenvalues). This leads first to expander graphs, then to Ramanujan graphs which are the best expanders from the spectral point of view. The name Ramanujan graphs was coined in 1986 by Lubotzky-Phillips-Sarnak as a sales pitch for the original construction of such graphs, that involved deep number theory: we will sketch that construction, that provides only graphs whose degree is of the form "1 + prime power". We will end with the probabilistic proof of Marcus-Spielman-Srivastava who showed in 2013 the existence of bipartite Ramanujan graphs of arbitrary degree (so that the name Ramanujan maybe is not so well-chosen, after all).

Petra Schwer: Bruhat–Tits buildings

This mini-course covers constructions of Bruhat-Tits buildings as well es some of their geometric and combinatorial properties. Buildings are geometric objects associated with reductive groups over non-archimedian local fields potentially with a discrete valuation. We will see explicit constructions of Bruhat-Tits buildings, axiomatic descriptions and the connection to BN-pairs. The most important geometric features of a building are its apartments and retractions onto apartments, which will also be introduced in the course.

Furthermore it will be explained how the geometry of retractions and the combinatorics of folded galleries provides a unified framework to study orbits in affine flag varieties. When \( X \) is a Bruhat-Tits building for a group over a local field, we can then relate labeled folded galleries and shadows to double coset intersections in affine flag varieties. This will allow us to hint at connections with the Hecke algebras discussed in Anne-Marie Aubert's course.

Indira Chatterji : Kazhdan's property (T)

Kazhdan’s property (T) for a group \( G \) is the isolation of the trivial representation in the unitary dual of \( G \). The first lecture will be devoted to a general introduction to property (T), the second lecture will explore consequences of property (T), and in particular that finite quotients give a family of expanders. The last lecture will be devoted to explaining a proof of property (T) for \( \mathrm{SL}(n,\mathbb{R}) \) due to Hee Oh and using estimates on the decay of the coefficients for an irreducible unitary representation.

François Thilmany: Lattices in p-adic Lie groups

In these lectures, we will explore the construction and basic properties of lattices in p-adic Lie groups. The provisional plan is the following:

• In lecture 1, we start with the general theory of lattice subgroups in locally compact groups. We define ‘lattice’, and give examples of lattices in some familiar groups. This will lead us to a famous theorem of Borel and Harish Chandra, describing a general procedure to construct arithmetic lattices in Lie groups. We briefly discuss the quotient of the group by a lattice, and state a criterion to determine when it is compact. To the extent permitted by time, we will survey the fundamental rigidity and arithmeticity theorems of Margulis.
• Lecture 2 will be dedicated to the specific properties of lattices in simple p-adic groups. In particular, we show that such lattices are always cocompact, virtually torsion-free, and finitely presented. We discuss the action of a lattice on the Bruhat-Tits building of the p-adic group.
• The contents of lecture 3 will be shaped by the needs of the other courses. We briefly discuss the representation theory of arithmetic p-adic lattices and the implications for the spectral properties of their action on the building. The remaining time may be spent reviewing some concepts or discussing some of the exercices.

Anne-Marie Aubert: Local representation theory and Hecke algebras

The mini-course will start with basic facts on the representation theory of locally compact groups. Next, we will describe the decomposition into irreducible unitary representations of the quasi-regular representation of a p-adic group \( G \) on \( L^2(G/L) \), where \( L \) is a lattice in \( G \).

We will introduce various Hecke algebras attached to \( G \), in particular the spherical Hecke algebra and the Iwahori-spherical Hecke algebra. We will study the representations of these algebras, and will explicit their relation with the above mentioned decomposition.

We will also explain the use of these Hecke algebras in the understanding of complex representations of a p-adic group \( G \), notably in the classification of the irreducible representations of \( G \), and how they can be related to the Bruhat-Tits building of \( G \).

The current list of registered participants can be found here.

• The summer school takes place from Monday, May 22 (around 10am) until Friday, May 26 (around 1pm).
• There will be no subscription fee.
• We can offer (limited) financial support for young participants (upon approval).
• The summer school will take place at Ghent University, Campus "Ledeganck", Auditorium 6.
• We will have a reception on Monday evening and a conference dinner on Wednesday evening.
• We plan to have a social event on Wednesday afternoon; more details will follow later.
• We ask all participants (except the speakers) to book their own accommodation. Here is a list of suggested hotels, which we kept short so as to maximize the likelihood that many participants will share the same hotel/hostel.
• Warning: some participants have received an email from travellerpoint.org related to accommodation. This is a scam. Do not respond to these messages.
• Questions? Send us an email: "hdx (at) ugent.be".

Ghent University
Université Catholique de Louvain
FWO-FNRS EOS programme