- Lecturer
- Marc Burger
- Coordinator
- Luca De Rosa

Rigidity Theory is a set of techniques invented to understand the structure of a certain class of discrete subgroups of Lie groups called lattices. Rigidity Theory is now being used in more general contexts like rigidity questions in geometric group theory on one hand and in higher Teichmueller theories on the other hand.

The aim of this course is to provide the main tools for Margulis' arithmeticity theorem. While the study of lattices in \( SL(2,\mathbb{R}) \) amounts to the study of complete hyperbolic surfaces with finite area and hence these lattices are organized into families with continuous parameters called Teichmueller spaces, Margulis' arithmeticity theorem says that for \( n \geq 3 \) every lattice in \( SL(n, \mathbb{R}) \) is obtained by an arithmetic construction, \( SL(n, \mathbb{Z}) \) being the simplest example thereof.

The main step in the proof of the arithmeticity theorem is the superrigidity theorem for linear representations of a lattice \(\Gamma\) in \( SL(n, \mathbb{R}) \). It says that if \({n \geq 3}\) and \( \pi : \Gamma \to GL(V) \) is an irreducible linear representation such that the corresponding \(\Gamma\)-action on the projective space \(\mathbb{P}(V)\) satisfies a certain dynamical condition (proximality), then \(\pi\) extends continuously to \( SL(n, \mathbb{R}) \). It is called superrigidity because it implies the famous Mostow-Prasad rigidity theorem that says that an abstract isomorphism between lattices in \( SL(n, \mathbb{R}) \), \( n \geq 3 \), extends to a Lie group automorphism of \( SL(n, \mathbb{R}) \), very much in contrast to the case of \( SL(2, \mathbb{R}) \).

The course will be organized as follows. We start discussing quasi-invariant measures on homogeneous spaces of locally compact groups, this will among other things clarify the notion of lattice. As a consequence of the Siegel-Weil formula, a game involving various invariant measures on various homogeneous spaces, we will show that \( SL(n, \mathbb{Z}) \) is a lattice in \( SL(n, \mathbb{R}) \). Using the fundamental theorem of geometry of numbers (Minkowski) we will also show that quaternion algebras over \( \mathbb{Q} \) lead to lattices in \( SL(2, \mathbb{R}) \). The bulk of the course is developing the tools for the proof of superrigidity. The main themes will be:

- Basic ergodic theory of group actions. The Howe-Moore theorem concerning the behaviour of coefficients of unitary representations of \( SL(n, \mathbb{R}) \) and its implications for ergodic theory.
- Furstenberg's boundary theory, in particular its use in the study of stationary measures on projective spaces \( \mathbb{P}(V) \) on which the lattice \( \Gamma \) acts. The relation between harmonic functions on \( SL(n, \mathbb{R}) \) and harmonic functions on the lattice \( \Gamma \).

You are encouraged to ask questions and start discussions on the Forum .

Some basic knowledge in Lie theory is useful but not vital. However basics of functional analysis and measure theory on locally compact spaces is essential.

- M.Burger: Handwritten notes of the course will be posted every week.
- Y. Benoist: "Five lectures on lattices in semisimple Lie groups", available on his homepage;
- G.A. Margulis: "Discrete Subgroups of semisimple Lie groups", Ergebnisse der Mathematik 3. folge, Band 17, Springer 1991.
- D. Witte-Morris: "Introduction to Arithmetic groups", available on Arxiv;
- R. Zimmer: "Ergodic Theory and Semisimple groups", Birkhauser 1984;