## About

This seminar is intended as a continuation of a joint seminar series, originally called "Seminar on geometric group theory", initiated by the mathematical departments of the universities Karlsruhe, Münster and Regensburg.

The core of this block seminar are the rigidity theorems of Margulis and Mostow. The participants shall be introduced to methods and ideas behind the proof of Margulis Superrigidity as well as to general background material on arithmetic groups, roughly based on the book "Introduction to arithmetic groups" by Dave Witte Morris. Some applications of both rigidity theorems will be given and further rigidity results - with particular focus on Monod's superrigidity for irreducible lattices - shall be discussed. The programme (see below) as well as relevant references are set up by the organisers, the 60-minutes talks shall be given by the participants themselves, sorted out accordingly to their suggestions.

This block seminar is meant to be as self-contained as possible. In particular, no background knowledge about arithmetic groups and the above-mentioned rigidity theorems is needed.

If you would like to participate, please **register by the 31st of January **(see below). If you volunteer to give a talk, please suggest three of the proposed talks which you would like to hold. At the beginning of February, we will distribute the talks according to the preferences.

A **detailed programme** with all the talks and the list of references can be found here.

Here you can find the homepages of the past seminars:

2012, Münster

#### Wednesday, March 21:

##### Talk 1: Introduction (tba)

##### Talk 2: Background on Lie groups (tba)

##### Talk 3: Geometric Motivation (tba)

##### Talk 4: Lattices (tba)

##### Talk 5: Arithmetic groups I (tba)

##### Talk 6: Arithmetic groups II (tba)

##### Discussion Session (-)

#### Thursday, March 22:

##### Talk 7: Ergodic theory (tba)

##### Talk 8: Amenable groups (tba)

##### Talk 9: Algebraic groups, actions on measures, and rationality of maps (tba)

##### Talk 10: Margulis Superrigidity I (tba)

##### Talk 11: Margulis Superrigidity II (tba)

##### Discussion Session (-)

#### Friday, March 23:

##### Talk 12: Mostow Rigidity (tba)

##### Talk 13: Monod’s Superrigidity Theorem (tba)

##### Talk 14: tba (tba)

### Talk 1: Introduction

This short talk will be given by one of the organisers. He will shortly present an outline of the seminar’s programme and summarise how the different topics are connected and will be combined for the proof of the Margulis Superrigidity Theorem.

### Talk 2: Background on Lie groups

The speaker of this talk should introduce basic concepts, examples and possibly some well-known facts about Lie groups. A good, short reference for the purposes of this seminar is [Mor15, Appendix A]. Define what a (linear) real Lie group is. Give examples of both connected and non-connected as well as compact Lie groups (for geometric motivation). Give examples of abelian, soluble and unipotent Lie groups (for algebraic motivation). Present structural concepts such as simplicity, semi-simplicity, isogenies (and maybe topological simple connectedness and coverings). Introduce the Haar measure and comment on its existence and uniqueness for Lie groups. If time permits, you could mention semisimple, hyperbolic, elliptic and unipotent elements and then present some structural properties such as Jordan Decomposition, Engel’s theorem, tori (in the sense of [Mor15, Chapter 8]) and parabolic subgroups. Literature: For the necessary concepts for the seminar, see [Mor15, Appendix A]. For basics about Lie groups, see e.g. [Zil10]. Standard references include [Hel01, Bou02, Hal03].

### Talk 3: Geometric Motivation

The aim of this talk is to explain how arithmetic groups arise in the context of locally symmetric spaces. If you want to give it, you should be familiar with basic definitions from Riemannian Geometry. Explain (locally) symmetric spaces and give an idea of how to produce a symmetric space from a connected Lie group and vice versa. Motivate the definition of a lattice (see [Mor15, p. 14]). Finally, state the Mostow Rigidity Theorem as in [Mor15, Theorem 1.3.10]. Literature: You can follow [Mor15, Chapter 1]. For basics about Lie groups and symmetric spaces, see [Zil10] or [Hel01].

### Talk 4: Lattices

Give the definition of lattices in Lie groups following [Mor15, 4.1] and work out an example (for instance as in [Mor15, 1.3.7]). Introduce the notion of irreducibility of a lattice (see e.g. [Mor15, 4.3]) and give an example of an irreducible lattice in a product of non-compact simple Lie groups. A nice example can be found e.g. in [Zim84, 2.2.5]. Discuss one of your favourite properties of lattices and, if the time allows, give an idea of how to show it for the case of SLn(Z) ⊂ SLn(R). Here, for instance, there is an almost self-contained discussion of Selberg’s Lemma in [Mor15, 4.8]. Literature: [Mor15] and [Zim84].

### Talk 5: Arithmetic groups I

Talk 5: Arithmetic groups I The first goal of this talk is to introduce arithmetic groups as in [Mor15, Definition 5.1.19]. To do so, introduce the necessary definitions such as Q−subgroups (see [Mor15, Section 4.3]). Give some examples of arithmetic groups. Here you can use the ones discussed in [Mor15, Chapter 6] or discuss results which describe the structure of arithmetic subgroups in a fixed ambient Lie group as in [Mor15, 18.6.1] or [Mor15, 18.6.3]. Please communicate with the speaker of Talk 6. Literature: [Mor15] and [Zim84].

### Talk 6: Arithmetic groups II

Talk 6: Arithmetic groups II In this talk the notions of lattices and arithmetic groups should be connected by the Margulis Arithmeticity Theorem. Different versions of this theorem can be found in [Mor15, 5.2.1] and [Zim84, 6.1.2]. One should be aware of the slightly different notions of arithmeticity in those books. Give an idea of the proof of the arithmeticity theorem. Here one can derive arithmeticity by using superrigidity as in [Mor15, 16.3] or give an idea of the more self-contained proof in [Zim84, 6.1]. Please communicate with the speaker of Talk 5. Literature: [Mor15] and [Zim84].

### Talk 7: Ergodic theory

In this talk the basic notions of ergodic theory should be introduced. An excellent source for this purpose is the second chapter of [Zim84]. In order to prove Margulis Superrigidity it is necessary to speak about Moore’s ergodicity theorem [Zim84, Thm. 2.2.6]. Give a sketch of the proof of Moore’s result, which involves the vanishing theorem of matrix coefficients [Zim84, 2.2.20]. Literature: [Zim84, Chapter 2]. See also [Mor15, Sections 14.1 and 14.2].

### Talk 8: Amenable groups

This talk is meant to be a brief introduction into the theory of amenability. Define what an amenable group is. Mention or give an idea of why soluble groups are amenable. You can discuss some consequences of amenability. Present Fürstenberg’s Lemma as stated in [Mor15, 12.6.1], which will also be needed in the proof of superrigidity. The remaining time can be used either to give more examples, to present equivalent definitions of amenability or to talk about the historical motivation to introduce amenability. Literature: Various definitions of amenability, the Fürstenberg Lemma and basic examples can be found in [Mor15, Chapter 12] and [Zim84, Chapter 4]. (Remark: Fürstenberg’s lemma appears roughly as Proposition 4.3.9 in [Zim84] in the context of amenable actions, presented in [Zim84, Section 4.3].) A lot more details on amenability are contained in [Pat88]. For the historical background, see [Mor15, Remark 12.4.3] and [TW16].

### Talk 9: Algebraic groups, actions on measures, and rationality of maps

The goal of this talk is to introduce the third main theory involved in the proof of Margulis’ theorem. Since it is not as self-contained as the others, it can be presented in a “crash course” style, focusing on elucidating the concepts to be introduced rather than on proofs. The speaker of this talk should be familiar with the theory of linear algebraic groups or at least have a basic background on algebraic geometry. Some knowledge on measure theory can be helpful, though you might supplement this with [Zim84, Appendix], [Mor15, Appendix B.6] and references therein. Briefly define algebraic varieties and the Zariski topology, then recall the definition of a linear algebraic group. Give examples of varieties, including projective spaces, and examples of actions of algebraic groups on varieties, possibly listing some well-known facts. A good quick reference including all of this is the beginning of [Zim84, Section 3.1]. Then, present just enough material to state (and, if time permits, sketch the proof of) Chevalley’s theorem (Prop. 3.1.4 of [Zim84]), Borel’s density theorem (e.g. as in [Zim84, 3.2.5]) and Corollaries 3.2.17, 3.2.18 and 3.2.19 of [Zim84, Section 3.2]. Define rationality of morphisms between varieties and present Section 3.4 of [Zim84], focusing on Propositions 3.4.1, 3.4.2 and Theorem 3.4.4 (here, you can restrict all results to the case k = R). Literature: [Zim84, Sections 3.1, 3.2 and 3.4]. For the measure-theoretic part, see [Zim84, Appendix] and [Mor15, Appendix B.6]. A good short reference for the relevant concepts from algebraic geometry is [Spr98, Chapter 1]. A more complete source on linear algebraic groups is [Bor91].

### Talk 10: Margulis Superrigidity I

The goal of this talk and the next one is to present the proof of the Margulis Superrigidity Theorem for (real) connected semisimple Lie groups of real rank ≥ 2, as done in Chapter 5 of [Zim84], using the tools presented so far in the seminar. The proof makes use of ergodic theory, results on amenability, and measure-theoretic aspects of actions of algebraic groups. The key steps in the proof involve the construction of some equivariant measurable maps, for which the contents of Talks 7 and 8 (especially Moore’s ergodicity theorem and Fürstenberg’s lemma) are needed, and checking that such maps have properties discussed in Talk 9 (more specifically, Borel’s density theorem, Corollaries 3.2.18 and 3.2.19, and Theorem 3.4.4 will be used directly). As presented in [Zim84, Chapter 5], the proof can be thematically divided into four parts, namely: Lemma 5.1.3 and the observations that follow; “Step 1” [Zim84, p. 97] and its proof; statement of “Step 2” [Zim84, p. 97] and Lemmata 5.1.4 and 5.1.5; Lemmata 5.1.6 – 5.1.8 and the proof of “Step 2”. Since the fourth part is the longest and most technical, it is conceivable that the speaker of this talk could state the Superrigidity Theorem and present the first and second parts as well as (most of) the third part above. In any case, you should communicate with the speaker of Talk 11 in order to properly subdivide the steps of the proof. Literature: [Zim84, Chapter 5].

### Talk 11: Margulis Superrigidity II

(See also the instructions above for Talk 10.) As mentioned, the fourth part of the proof of Margulis’ theorem is lengthier and more technical. It consists of a series of lemmata involving tori and unipotent subgroups as well as the concluding proof of “Step 2”, so it might be at least as long as the first two parts of the proof. It is therefore conceivable that this talk focuses on it, recalling, as appropriate, the arguments already presented in the previous talk. The speakers of Talks 10 and 11 should, of course, coordinate among them on how the whole proof of the theorem is to be divided. After concluding the proof, and if time allows, the speaker of Talk 11 could either: Mention Margulis Superrigidity over complex and p-adic fields and relate all of those superrigidity theorems to arithmeticity as an application; mention or state Margulis superrigidity in its full generality [Mar91], briefly elucidating similarities and differences with the theory discussed in the seminar; point out her/his favourite application of the Margulis Superrigidity Theorem (see e.g. [Mor15, 16.2]). Literature: [Zim84, Chapter 5], [Mar91, Chapter VII], [Mor15, Chapter 16].

### Talk 12: Mostow Rigidity

Prove a variant of the Mostow Rigidity Theorem (in real rank ≥ 2) as an application of Margulis’ theorem, either as in [Zim84, Chapter 5] or working out the outline given in [Mor15, Chapter 15]. The remainder of the talk should be devoted to the geometric translation of the above mentioned version, that is, obtaining the classical geometric formulation of rigidity [Spa04] from the version of Mostow’s theorem proved above (with the appropriate modifications, if needed). If time permits you could discuss Mostow Rigidity from the point of view of quasi-isometries (see [Mor15, 15.4]). Literature: [Zim84, Chapter 5], [Mor15, Chapter 15 and Section 16.2] and [Spa04].

### Talk 13: Monod’s Superrigidity Theorem

This talk is about more recent results on rigidity which were proven by Nicolas Monod [Mon06]. The goal of this talk is to understand the differences and similarities of the rigidity results of Margulis and Monod. Introduce the basic notions of CAT(0) geometry (see e.g. [Mon06, Section 3]) which are necessary to formulate the rigidity result of Monod ([Mon06, Theorem 6]). Explain why Monod’s results generalise Margulis’ results [Mar91] in the case where the given Lie group decomposes into at least 2 factors and the lattice is uniform. To do so, explain where CAT(0) spaces appear in the context of Margulis’ rigidity theorem (Hint: Look at Lie group actions on symmetric spaces or Bruhat–Tits buildings, which are often CAT(0). See e.g. [BH99, II.10].) If time permits, give a rough idea on how to prove [Mon06, Theorem 6] and possibly elucidate why our version of superrigidity proved in the seminar does not necessarily fit the framework of Monod’s theorem (so that both results might be understood as mutually complementary). Literature: [Mon06], [BH99] and [Mar91].

### Talk 14: tba

The last talk will be given by a member of the faculty.