# Topics in Rigidity Theory Autumn 2020

Lecturer
Marc Burger
Coordinator
Luca De Rosa

## Content

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 $$\Gamma$$.

## Forum

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

## Prerequisites

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.

## Schedule

Lectures are taking place on Thursdays from 14 to 17, in HG G19.1 . Recordings of the lectures can be found on the video portal . The live streaming can be accessed here: Live streaming .
Lecture Time Content Notes
Lecture 1 September 17, 2020 Introduction, part 1: Definition of lattice. Motivational example: Classifying certain types of geometric structures on a fixed smooth surface leads to understanding lattices in $$SL(2,\mathbb{R})$$. Handwritten Notes 1
Lecture 2 September 24, 2020 Introduction, part 2: Mostow's rigidity theorem, definition of linear algebraic group, commensurable subgroups, arithmetic lattices. The arithmeticity theorem. Motivating question for the remaining of the course: to which extent are all lattices in simple Lie groups arithmetic? Handwritten Notes 2
Lecture 3 October 1, 2020 Introduction, part 3: Notions of commensurator of a subgroup, correspondence, strongly irreducible representation, proximal action. Statement of Margulis' Superrigidity theorem. Overview of how superrigidity implies arithmeticity thoerem, for $$G= SL (n, \mathbb{R})$$, $$n \geq 3$$. Chapter 2: Introduction, modular function, unimodular groups. Handwritten Notes 3
Lecture 4 October 8, 2020 Examples of Haar measures, existence theorem for semi-invariant positive Radon measure with a given modulus (Weil '38). Handwritten Notes 4
October 15, 2020 This lecture has been cancelled.
Lecture 5 October 22, 2020 End of chapter 2. Introduction on chapter 3: 'Examples of Arithmetic Lattices', Minkowski's theorem, beginning of the proof that $$SL(n, \mathbb{Z})$$ is a lattice in $$SL(n, \mathbb{R})$$. Handwritten Notes 5
Lecture 6 October 29, 2020 Completion of the proof that $$SL(n, \mathbb{Z})$$ is a lattice in $$SL(n, \mathbb{R})$$, introduction to chapter 4: 'Ergodic Actions and the Howe-Moore theorem', definition of ergodic action and first examples. Handwritten Notes 6
Lecture 7 November 5, 2020 Essentially constant and essentially R-invariant maps. Unitary representations and matrix coefficients. The Howe-Moore theorem, beginning of the proof for $$SL(n, \mathbb{R})$$. Handwritten Notes 7-8
Lecture 8 November 12, 2020 Proof of Howe-Moore theorem.
Lecture 9 November 19, 2020 Introduction to chapeter 5: 'Lattices and Boundary Maps', Poisson transform, $$\mu$$-harmonic maps and the mean value property, the Poisson boundary, description and overview of the main results of the chapter. Handwritten Notes 9-10
Lecture 10 November 26, 2020 Recap of the main results of the chapter, and strategy of the proofs. Equivariant maps into probability measures, Theorem 5.5 with proof.
Lecture 11 December 3, 2020 Overview of necessary steps to prove Superrigidity Theorem. Definition of martingale, the martigale's convergence theorem, and how it relates to bounded harmonic functions. How the martingale's convergence theorem fit in out goal: Proposition 5.15 with proof. Handwritten Notes 11
Lecture 12 December 10, 2020 Quasi-projective maps, contracting sequences in $$GL(n. V)$$. Proof of Theorem 5.16. Handwritten Notes 12

Proof of Thm 5.13 from Margulis' Book
Lecture 13 December 17, 2020 Chapter 6: 'The Superrigidity Theorem', with proof. Conclusions. Handwritten Notes 13, part 1

Handwritten Notes 13, part 2

## Literature

• 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;