Oberseminar Algebra und Geometrie

Sommersemester 2018: Arithmetic Groups - Basics and Selected Applications

Organised by I. Halupczok, B. Klopsch, S. Schröer and M. Zibrowius.

All talks take place Fridays at 12:30 in 25.22.03.73.

Short description

The aim of the seminar is to get to know arithmetic groups. We will mainly follow the self-contained Lecture Notes [Hu80] of James E. Humphreys. In particular, we cover: the necessary number theoretic background in the setting of locally compact abelian groups and discrete subgroups; the general linear and special linear groups, with an emphasis on “reduction theory” (computation of fundamental domains; information about finite generation and finite presentations); the Congruence Subgroup Problem. A few other sources, including the books [Mo15] by Morris Witte and [PR94] by Platonov and Rapinchuk, are listed below. Due to time considerations, we will not prove all, but only selected results. In particular, the last two talks are meant to survey technical results rather than to treat them in detail.

Schedule

13.04.18 1.  Locally compact groups and fields (Florian Severin)

Main source: Ch. I, Sec 1-3 of [Hu80]

Keywords: Haar measure, module of an automorphism, local and global fields, classification and strucutre theorems, adele ring of a global field.
20.04.18 2.  The additive group (Kevin Langlois)

Main source: Ch. II, Sec 4-6 of [Hu80]

Keywords: The quotient of the adele group by the number field, fundamental domains and product formula, volume of a fundamental domain, Strong Approximation Theorem.
27.04.18 3.  The multiplicative group & Example: The modular group (Johannes Fischer)

Main source: Ch III, Sec 7,8 and Ch IV, Sec 9 of [Hu80]

Keywords: Ideles, compactness theorem, class numbers and units, the modular group.
04.05.18 4.  GLn and SLn over R: Siegel sets and Applications (Moritz Petschick)

Main source: Ch IV, Sec 10,11 of [Hu80]

Keywords: Siegel sets in GL(n,R), Iwasawa decomposition, Minimum Principle, Siegel sets in SL(n,R), reduction of positive definite quadratic forms.
11.05.18 5.  GLn and SLn over R: BN-pairs (Kathika Rajeev)

Main source: Ch IV, Sec 12 of [Hu80]

Keywords: BN-pairs, axioms and Bruhat decomposition, parabolic subgroups, complements.
18.05.18 6.  GLn and SLn over R: Siegel property and applications (Andrea Fanelli)

Main source: Ch IV, Sec 13 of [Hu80]

Keywords: Siegel sets revisited, fundamental sets and Siegel property, Harish-Chandra's theorem, finite presentations, corners.
25.05.18 7.  GLn and SLn: Adelic groups (Peter Arndt)

Main source: Ch V, Sec 14 of [Hu80]

Keywords: Adelisation of a linear group, class number, Strong Approximation, reduction theory.
01.06.18 8.  GLn and SL_n: SL2 over p-adic fields (David Bradley-Williams)

Main source: Ch V, Sec 15 of [Hu80]

Keywords: Infinite dihedral group, lattices in K2, BN-pairs, building attached to BN-pair, Ihara's theorem, maximal compact subgroups.
08.06.18 9.  The congruence subgroup problem (Benjamin Klopsch)

Main source: Ch VI, Sec 16 of [Hu80]; see also Ch II, Sec 1-3 of [NKV11]

Keywords: Subgroup topologies, profinite groups, completions, congruence kernel.
15.06.18 10.  Interlude: Strong approximation methods in group theory (Matteo Vannacci)

Main source: Ch II, Sec 4-7 of [KNV11]

Keywords: Strong Approxiamtion Theorem, Lubotzky's Alternative, Nori-Weisfeiler theorem.
22.06.18 11.  The congruence kernel of SL(n,Z) (Benno Kuckuck)

Main source: Ch VI, Sec 17 of [Hu80]

Keywords: Conguence subgroups, q-elementary subgroups, congruence kernel of SL(n,Z).
29.06.18 final day of the Workshop: Trees, dynamics and locally compact groups at HHU
(organised by Alejandra Garrido and Benjamin Klopsch)
06.07.18 12.  The Steinberg group (Marcus Zibrowius)

Main source: Ch VI, Sec 18 of [Hu80]; compare last semester's Talk 7

Keywords: Generators and relations, upper unitriangular group, monomial group, Steinberg symbols, universal property.
13.07.18 13.  Survey: Matsumoto's theorem and Moore's theory (Oihana Garaialde)

Main source: Ch VI, Sec 19, 20 of [Hu80]

Keywords: Central extensions and cocycles, Matsumoto's theorem, topological Steinber symbols, local and global theorems, central extensions, fundamental group, the congruence kernel revisited.
20.07.18 14.  Survey: Rigidity theorems (Alejandra Garrido)

Main source: Ch 15, 16 of [Mo15]

Keywords: Mostow Rigidity Theorem, quasi-isometric rigidity, Margulis Superrigidity Theorem.
Programme discussion for the next semester (chaired by Stefan Schröer)

Literature

[Bo69]  Borel, Armand: Introduction aux groupes arithmetiques, Publications de l'Institut de Mathematique de l'Universite de Strasbourg, Hermann, Paris 1969
[Hu80]  Humphreys, James E.: Arithmetic groups, Lecture Notes in Mathematics 789, Springer, Berlin, 1980.
[Ji08]  Ji, Lizhen: Arithmetic groups and their generalizations. What, why, and how, AMS/IP Studies in Advanced Mathematics 43, International Press, Cambridge, MA, 2008.
[KNV11]  Klopsch, Benjamin; Nikolov, Nikolay; Voll, Christopher: Lectures on profinite topics in group theory (Dan Segal ed.), London Mathematical Society Student Texts 77, Cambridge University Press, Cambridge, 2011.
[Mo15] (Witte) Morris, Dave: Introduction to arithmetic groups, Deductive Press, 2015.
[PR94] Platonov, Vladimir; Rapinchuk, Andrei: Algebraic groups and number theory, Pure and Applied Mathematics 139, Academic Press, Inc., Boston, MA, 1994.
[P04] Prasad, Gopal: Borel's contributions to arithmetic groups and their cohomology, Gaz. Math. 102 (2004), 15-24. [article]


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