# Winter term 2021/22: Superrigidity

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

Organised by H. Kammeyer

All talks take place on Fridays at 12:30 pm in 25.22.03.73. The "3G" rule applies. Talks should be kept to a maximum of 90 minutes. Please ensure to be on the mailing list to be notified about short-term changes. If at any time during the semester, new university regulations should rule out meetings in presence, the seminar will be transferred to an online format.

## Aims and Content - a short description

The group $\operatorname{SL}_2(\mathbb{R})$ contains the free group $F_2$ as a lattice (meaning a discrete subgroup with finite covolume) and of course a free group has a continuous variety of representations. This is fundamentally different for a lattice $\Gamma \subset \operatorname{SL}_3(\mathbb{R})$. One can obtain a representation of $\Gamma$ by restricting a representation of the Lie group $\operatorname{SL}_3(\mathbb{R})$ but that is essentially the only way to find a representation of the abstract group $\Gamma$ over a local field. It is a peculiar fact (and a Fields medal winning theorem) that the "higher rank" assumption effects the rigidity property that lattices behave as if they were "discrete copies" of the surrounding Lie groups. The proof uses a stunning blend of techniques from Lie theory, algebraic and arithmetic groups, representation theory and ergodic theory. As another reward, the superrigidity theorem not only says that the representation theory of higher rank lattices is rigid but also the way to come up with them is limited: they are all arithmetic in the sense that they are essentially given by the integral points of an algebraic group.

## Schedule

 15.10.21 get ready - no seminar yet 22.10.21 (For once at 02:30 pm!) 1.  Construction of lattices (Pablo Cubides Kovacsics) Chapter 1 in [B] The purpose of this talk is to show that arithmetic subgroups of orthogonal groups are lattices. The real purpose of this talk is to see in a particular example how methods from ergodic theory, number theory, and group theory combine to prove a powerful theorem, as will be the general theme of this seminar. Keywords: lattices in locally compact groups, Haar measure, unimodular, HC hypothesis, recurrence of random walks, G-invariant Radon mesures are finite and stationary, functions contracted by convolution, Mahler criterion, Minkowski's theorem, construction of a function verifying HC, conclusion. 29.10.21 2.  Semisimple Lie algebras (Max Lindh) Chapter 2 in [B] A recap of basic concepts in the theory of semisimple Lie algebras. Keywords: Lie algebras, ideals, nilpotent Lie algebras, Theorems of Engel and Lie, Killing form criterion for semisimplicity, Jordan decomposition, $\mathfrak{sl}_2$ representations, $\mathfrak{sl}_2$-triples, root systems. 05.11.21 3.  Semisimple Lie groups (Arthur Martirosian) Chapter 3 in [B] Semisimple Lie groups and their decompositions. Keywords: Equivalence of categories of compact center-free Lie groups and complex semisimple Lie algebras, real forms of complex semisimple Lie algebras, Cartan involutions, Cartan subalgebras and subspaces, real root space decomposition, Cartan and Iwasawa decompositions, parabolic subgroups. 12.11.21 4.  Algebraic groups (Margherita Piccolo) Chapter 4 in [B] An introduction to algebraic groups from a classical viewpoint. Keywords: elementary vocabulary on algebraic varieties, Chevalley's theorem on regular morphisms, linear algebraic groups, algebraic actions, realization of subgroups as stabilizers of projectivized representations, semisimple and unipotent elements, vocabulary on algebraic groups. 19.11.21 5.  Arithmetic groups (Iker de las Heras) Chapter 5 in [B] The generalization of the first talk to the Borel--Harish-Chandra theorem, stating that arithmetic groups in algebraic groups witout rational characters are lattices. Keywords: definition of arithmetic groups, theorem of Borel--Harish-Chandra, Godement criterion, embedding of the coset space of an arithmetic group into the space of lattices, proof of the Borel--Harish-Chandra theorem in the isotropic and anisotropic case, first for simple groups, then for reductive groups, then in the general case. 26.11.21 6.  Mixing properties (free) Chapter 6 in [B] As a consequence of the Howe--More theorem on matrix coefficients of unitary representations, we show that a simple Lie group acts mixingly on the set of cosets of a lattice. Keywords: unitary representations and matrix coefficients, Howe--More theorem, conclusion on fixed vectors, Mautner's lemma and the $\mathrm{SL}_2(\mathbb{R})$ case conclusion of mixing property, proof of the Howe--Moore theorem. Only explain the statement of the Duke--Rudnick--Sarnak theorem. Giving an indication of the proof is optional. 03.12.21 7.  Lattices (free) Chapter 7 in [B] We study concepts and properties of lattices in semisimple Lie groups. Keywords: Borel density theorem, irreducibility, amenable groups, the equivariant Furstenberg boundary map. 10.12.21 8.  Ergodic theory (Moritz Petschick) Chapter 8 in [B] We introduce and study basic concepts of ergodic theory. Keywords: ergodicity, dense orbits, Birkhoff ergodicity theorem, martingales, Doob's martingale convergence theorem, stopping times. 17.12.21 9.  Stationary measures (free) Chapter 9 in [B] Stationary measures describe the asymptotic behavior of random walks. We study their existence and uniqueness and deduce existence and uniqueness properties of boundary maps. Keywords: stationary measures, existence and random walk interpretation, proximality and uniqueness of stationary measures and boundary maps, stationary measures on the boundary which are invariant under a maximal compact subgroup. 07.01.22 10. Superrigidity (free) Chapter 10 in [B] In this talk we put the pieces together and prove the Margulis superrigidity theorem. Keywords: statement of the theorem, meaning in the p-adic case, necessity of hypotheses, proof strategy, reduction to the proximal case, construction of boundary map, finite dimensionality of the space of translates of the boundary map (stress that it is at this point that the higher rank assumption enters), extension of the homomorphism from the lattice to the Lie group. 14.01.22 11.  Arithmeticity (free) Chapter 11 in [B] We conclude the Margulis arithmeticity theorem from the p-adic and real versions of the superrigidity theorem. Keywords: finite generation of lattices, algebraicity of eigenvalues, the trace field is a number field, restriction of scalars. 21.01.22 12.  Free slot (free) 28.01.22 13.  Free slot (free) 04.02.22 14.  Discussion for next semester's seminar

## Literature

We will follow the lecture notes by Yves Benoist:
 [B] Benoist, Y.: Réseaux des groupes de Lie available here.

Parts of the course, in particular on arithmetic groups and boundaries, are also treated in
 [B2] Benoist, Y.: Five lectures on lattices in semisimple Lie groups available here

In addition, you are encouraged to consult the following English sources:

On superrigidity:
 [M] Margulis, G.: Discrete Subgroups of Semisimple Lie Groups, Springer 1991 [Z] Zimmer, R. J.: Ergodic Theory and Semisimple Groups, Birkhäuser 1984

On Lie theory:
 [Hu1] Humphreys, J.: Introduction to Lie Algebras and Representation Theory, Springer 1972 [He] Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces, AMS 2001 [Se] Serre, J.P.: Complex Semisimple Lie Algebras, Springer 2001

On algebraic groups:
 [Sp] Springer, T.: Linear Algebraic Groups, Birkhäuser 1998 [Hu2] Humphreys, J.: Linear Algebraic Groups, Springer 1975

On arithmetic groups:
 [R] Raghunathan, M.S.: Discrete Subgroups of Lie Groups, Springer 1972 [WM] Witte Morris, D.: Introduction to Arithmetic Groups

## Archive

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)