Discrete subgroups of semisimple lie groups pdf download

Discrete subgroups of solvable lie groups have been fairly thoroughly studied, but the results are less complete than those obtained for nilpotent groups. On normal subgroups of semisimple lie groups on normal subgroups of semisimple lie groups george michael, a. Mostow notes by gopal prasad no part of this book may be reproduced in any form by print, micro. We introduce and prove the equivalence of several conditions, capturing rank one behavior of discrete subgroups of higher rank lie groups.

Discrete subgroups of semisimple lie groups ergebnisse. Click download or read online button to get theory of lie groups book now. For a large part, they summarise relevant material from knapps book 12. A geometric construction of the discrete series for semisimple lie groups 3 k local integrability of the harishchandra characters. Then 2 is unimodular, and hh has a finite z invariant measure. Discrete subgroups of real semisimple lie groups 557 let us introduce a topology into the space w, consisting of all lattices lying in some finitedimensional euclidean space r. Maximal operators associated to discrete subgroups.

Tata institute of fundamental research, bombay 1969. Various types of discrete subgroups of lie groups arise in the theory of functions of complex variables, arithmetic, geometry, and crystallography. Since the notion of lie group is sufficiently general, the author not only proves results in the classical geometry setting, but also obtains theorems of an algebraic nature, e. Let g be a topological group and f an arbitrary group. Then r acts without fixed points on the left on the symmetric space x gk, and can therefore be identified with the fundamental. Ix t 7 dco o oe d represents the contribution of the discrete series to the plancherel formula of g, we intend to obtain explicit formulas. Irreducible characters of semisimple lie groups ii. Discrete subgroups of real semisimple lie groups g a margulison some groups of motions of noncompact nonsingular symmetric spaces of rank 1 m e novodvorskistructure of topological locally projectivelynilpotent groups, and of groups with a normalizer condition v p platonovrecent citations lower bound for the volumes of quaternionic hyperbolic. Let g and b be the lie algebras of g and b respectively.

Let g be a semisimple real algebraic lie group of real rank at least 2, and let u be the unipotent radical of a nontrivial parabolic subgroup. Suppose further that g is linear and that r contains no elements of finite order. For purely formal reasons, jfkogk, f can also be described as the kth relative lie algebra cohomology group of f. Margulis wrote in the preface of his book discrete subgroups of semisimple lie groups that a number of important topics have been omitted. Rigidity of furstenberg entropy for semisimple lie group. They are direct generalizations of rank one equivalents to convex cocompactness. Semisimple lie groups 79 regardless of the particular nature of f, fkgk, f contains at least fgk, f, i.

Arithmetic properties of discrete subgroups iopscience. Lie groups and lie algebras ii discrete subgroups of lie. These formulas are used to give a simple formula for the fourier transform of orbital integrals of regular semisimple orbits. Orbits of oneparameter groups i plays in a lie algebra. Nondiscrete uniform subgroups of semisimple lie groups. Discrete and continuous cosine transform generalized to. Discrete subgroups generated by lattices in opposite. The first part of this book on discrete subgroups of lie groups is written by e. A geometric construction of the discrete series for. Maximal operators associated to discrete subgroups of nilpotent lie groups by akos magyar, elias m. Contemporary mathematics volume cocompact subgroups of.

Lie algebras are an essential tool in studying both algebraic groups and lie groups. Characteristic free measure rigidity for the action of. In the special case of subgroups of r n, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all. We show that certain discrete subgroups of semisimple lie groups satisfy rigidity properties and that a subclass of these discrete groups are actually of finite covolume. We develop and describe continuous and discrete transforms of class functions on compact semisimple lie group g as their expansions into series of uncommon special functions, called here cfunctions in recognition of the fact that the functions generalize cosine to any dimension n download bok. Introduction and statement of main theorem the purpose of this paper is to prove a maximal theorem for averages taken over suitable discrete subvarieties of nilpotent lie groups. The point of this paper is to study the low dimensional cohomology theory of ergodic actions of semisimple lie groups and their lattice subgroups. In lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. Theory of lie groups download ebook pdf, epub, tuebl, mobi. Semisimple lie groups and discrete subgroups by robert j. Chapter i develops the basic theory of lie algebras, including the fundamental theorems of engel, lie, cartan, weyl, ado, and poincarebirkhoffwitt.

Raghunathan and others published discrete subgroups of lie groups find, read and cite all the research you need on researchgate. On the first cohomology of discrete subgroups of semisimple. For these reasons it is important to study the subgroup structure of the almost simple groups, and in particular their maximal subgroups. Cocompact subgroups of semisimple lie groups lemma 1. Raghunathan received june 19, 1978 the paper referred to in the title appeared in this journal in 1966 vol. Of these i follows from the fact that h appears as a subrepresentation of an induced representation see the simple proof by casselman 9. In this article those discrete subgroups of the group g of real unimodular matrices of order three are investigated which have the property that the factor space of the group g by them has finite volume and is not compact.

The rigidity theorem for ergodic actions of semisimple lie groups of rrank at least 2 27 shows in one. Give different characterizations of the subclass of anosov subgroups, which generalize convexcocompact subgroups of rank 1 lie groups, in terms of various equivalent dynamical and geometric properties such as. We prove that a discrete zariskidense subgroup of g that contains an irreducible lattice of u is an arithmetic lattice of g. In particular, under some rather weak assumptions on a semisimple lie group g we prove that every discrete subgroup of g with a noncompact factor space of finite volume that satisfies some natural irreducibility conditions, is an arithmetic subgroup of g. The fundamental result is formulated in the introduction. Volume invariant and maximal representations of discrete. Vanishing theorems for lie algebra cohomology and the cohomology of discrete subgroups of semisimple lie groups wilfried schmid department of mathematics, harvard university, cambridge, massachusetts 028 1. Our results can be applied to the theory of algebraic groups over global fields. Unfortunately theorem 1 of the paper as it stands is incorrect.

Discrete subgroups of semisimple lie groups gregori a. Any lattice in a solvable lie group is a uniform discrete subgroup. Let 2 be a locally compact group, with a closed, unimodular, cocompact subgroup h. Let be the free group on generators, the outer automorphism group is the most important group in combinatorial group theory. A major achievement in the theory of discrete subgroups of semisimple lie groups is margulis superrigidity theorem. The most significant of these is the theory of kleinian groups and thurstons theory of 3dimensional manifolds. Vanishing theorems for cohomology groups associated to discrete subgroups of semisimple lie groups. The discrete series of semisimple groups peter hochs september 5, 2019 abstract these notes contain some basic facts about discrete series representations of semisimple lie groups. For a thorough treatment of the history of the subject, see. Relativizing characterizations of anosov subgroups, i.

In mathematics, a simple lie group is a connected nonabelian lie group g which does not have nontrivial connected normal subgroups together with the commutative lie group of the real numbers, and that of the unitmagnitude complex numbers, u1 the unit circle, simple lie groups give the atomic blocks that make up all finitedimensional connected lie groups via the operation of. This solves a conjecture of margulis and extends previous works of selberg and oh. Lattices are best thought of as discrete approximations of continuous groups such as lie groups. Introduction let g be a connected semisimple lie group with finite center, fc g a. Analogs of wieners ergodic theorems for semisimple lie groups. It is assumed that the reader has considerable familiarity with lie groups and algebraic groups. Margulis, discrete subgroups of semisimple lie groups, springerverlag, page, a subgroup h of an algebraic group g is called algebraic if h is an algebraic subvariety of g. Sungwoon kim, inkang kim submitted on 22 may 2012 v1, last revised 21 sep 2012 this version, v2.

However, i do not believe one can use this technique to construct simple discrete subgroups of lie groups. Explicit formulas for discrete series characters on noncompact cartan subgroups are given. Discrete subgroups of lie groups and discrete transformation. Mapping class groups of surfaces with the actions on the teichmuller spaces. We relativize various dynamical and coarse geometric characterizations of anosov subgroups given in our earlier work, extending the class from intrinsically hyperbolic to. We start with the definitions of borel subgroups and subalgebras, parabolic subgroups and subalgebras, and complex flag manifolds. On the first cohomology of discrete subgroups of semi. Vanishing theorems for lie algebra cohomology and the. On the maximal solvable subgroups of semisimple algebraic. By lie groups we not only mean real lie groups, but also the sets of krational points of algebraic groups over local fields k and their direct products. For example, it is intuitively clear that the subgroup of integer vectors looks like the real vector space in some sense, while both groups are essentially different.

In this paper we also study various results from the theory of algebraic groups and their. Mar 07, 2020 for these reasons it is important to study the subgroup structure of the almost simple groups, and in particular their maximal subgroups. We propose several common extensions of the classes of anosov subgroups and geometrically finite kleinian groups among discrete subgroups of semisimple lie groups. Raghunathandiscrete groups and qstructures on semisimple lie groups proc. If h is a lie group, and h,rad h e, then h is unimodular. Finite simple subgroups of semisimple complex lie groups. Zimmer, on the cohomology of ergodic actions of semisimple lie groups and discrete subgroups, amer. This site is like a library, use search box in the widget to get ebook that you want. On discrete subgroups of lie groups by andrt weil received february 1, 1960 1. Discrete subgroups of real semisimple lie groups iopscience. G of higher rank semisimple lie groups, which exhibit some rank 1 behavior. Our interest, by and large, is in a special class of discrete subgroups of lie groups, viz.

This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semisimple lie groups. By looking at these examples i realized what i should have known long time ago, namely that infinite permutation group with dynkin diagram infinite line with integer nodes is virtually simple. We first establish some properties of the fundamental invariant associated with a g. On normal subgroups of semisimple lie groups, results in. Ergodic theory, group representations, and rigidity. We consider the action of a semisimple lie group g on a compact manifold and more generally a borel space x, with a measure. We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple lie groups. Readings introduction to lie groups mathematics mit. Besides discrete subgroups of lie groups, two other very important discrete transformation groups are. Geometry and topology of coadjoint orbits of semisimple lie groups bernatska, julia and holod, petro, 2008. To do this we shall say that a sequence of lattices converges to a lattice. Journal of lie theory volume 22 2012 11691179 c 2012 heldermann verlag on the maximal solvable subgroups of semisimple algebraic groups hassan azad, indranil biswas, and pralay chatterjee communicated by e. Volume invariant and maximal representations of discrete subgroups of lie groups authors.