We relativize various dynamical and coarse geometric characterizations of anosov subgroups given in our earlier work, extending the class from intrinsically hyperbolic to. Nondiscrete uniform subgroups of semisimple lie groups. The fundamental result is formulated in the introduction. Discrete subgroups of semisimple lie groups ergebnisse. We start with the definitions of borel subgroups and subalgebras, parabolic subgroups and subalgebras, and complex flag manifolds. Mostow notes by gopal prasad no part of this book may be reproduced in any form by print, micro. Lie algebras are an essential tool in studying both algebraic groups and lie groups. If h is a lie group, and h,rad h e, then h is unimodular. Discrete subgroups of solvable lie groups have been fairly thoroughly studied, but the results are less complete than those obtained for nilpotent groups. Characteristic free measure rigidity for the action of. Let g be a topological group and f an arbitrary group. Various types of discrete subgroups of lie groups arise in the theory of functions of complex variables, arithmetic, geometry, and crystallography.
Contemporary mathematics volume cocompact subgroups of. Rigidity of furstenberg entropy for semisimple lie group. Besides discrete subgroups of lie groups, two other very important discrete transformation groups are. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semisimple lie groups. Any lattice in a solvable lie group is a uniform discrete subgroup. For purely formal reasons, jfkogk, f can also be described as the kth relative lie algebra cohomology group of f. Tata institute of fundamental research, bombay 1969. 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.
Chapter i develops the basic theory of lie algebras, including the fundamental theorems of engel, lie, cartan, weyl, ado, and poincarebirkhoffwitt. Then r acts without fixed points on the left on the symmetric space x gk, and can therefore be identified with the fundamental. A major achievement in the theory of discrete subgroups of semisimple lie groups is margulis superrigidity theorem. For a thorough treatment of the history of the subject, see. On the first cohomology of discrete subgroups of semisimple. We prove that a discrete zariskidense subgroup of g that contains an irreducible lattice of u is an arithmetic lattice of g.
G of higher rank semisimple lie groups, which exhibit some rank 1 behavior. Then 2 is unimodular, and hh has a finite z invariant measure. Raghunathan received june 19, 1978 the paper referred to in the title appeared in this journal in 1966 vol. Unfortunately theorem 1 of the paper as it stands is incorrect. It is assumed that the reader has considerable familiarity with lie groups and algebraic groups. We propose several common extensions of the classes of anosov subgroups and geometrically finite kleinian groups among discrete subgroups of semisimple lie groups. On the maximal solvable subgroups of semisimple algebraic. We first establish some properties of the fundamental invariant associated with a g. A geometric construction of the discrete series for. On normal subgroups of semisimple lie groups on normal subgroups of semisimple lie groups george michael, a. Raghunathandiscrete groups and qstructures on semisimple lie groups proc. 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. 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.
Ergodic theory, group representations, and rigidity. However, i do not believe one can use this technique to construct simple discrete subgroups of lie groups. 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. Volume invariant and maximal representations of discrete subgroups of lie groups authors. 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. Geometry and topology of coadjoint orbits of semisimple lie groups bernatska, julia and holod, petro, 2008. Let g and b be the lie algebras of g and b respectively. Discrete subgroups of lie groups and discrete transformation. On normal subgroups of semisimple lie groups, results in. Click download or read online button to get theory of lie groups book now. Maximal operators associated to discrete subgroups.
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. 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. Analogs of wieners ergodic theorems for semisimple lie groups. Mapping class groups of surfaces with the actions on the teichmuller spaces. Lie groups and lie algebras ii discrete subgroups of lie. 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. Theory of lie groups download ebook pdf, epub, tuebl, mobi. For these reasons it is important to study the subgroup structure of the almost simple groups, and in particular their maximal subgroups. Introduction let g be a connected semisimple lie group with finite center, fc g a.
Let 2 be a locally compact group, with a closed, unimodular, cocompact subgroup h. Discrete subgroups of semisimple lie groups gregori a. To do this we shall say that a sequence of lattices converges to a lattice. 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. Of these i follows from the fact that h appears as a subrepresentation of an induced representation see the simple proof by casselman 9. 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. Let be the free group on generators, the outer automorphism group is the most important group in combinatorial group theory. 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. Cocompact subgroups of semisimple lie groups lemma 1. 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. The point of this paper is to study the low dimensional cohomology theory of ergodic actions of semisimple lie groups and their lattice subgroups. Sungwoon kim, inkang kim submitted on 22 may 2012 v1, last revised 21 sep 2012 this version, v2. 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.
These formulas are used to give a simple formula for the fourier transform of orbital integrals of regular semisimple orbits. The rigidity theorem for ergodic actions of semisimple lie groups of rrank at least 2 27 shows in one. Lattices are best thought of as discrete approximations of continuous groups such as lie groups. Margulis wrote in the preface of his book discrete subgroups of semisimple lie groups that a number of important topics have been omitted. Suppose further that g is linear and that r contains no elements of finite order. Explicit formulas for discrete series characters on noncompact cartan subgroups are given. Orbits of oneparameter groups i plays in a lie algebra. 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. A geometric construction of the discrete series for semisimple lie groups 3 k local integrability of the harishchandra characters. They are direct generalizations of rank one equivalents to convex cocompactness. 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. Relativizing characterizations of anosov subgroups, i. 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.
For a large part, they summarise relevant material from knapps book 12. Arithmetic properties of discrete subgroups iopscience. In this paper we also study various results from the theory of algebraic groups and their. 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. Volume invariant and maximal representations of discrete. Discrete subgroups of real semisimple lie groups iopscience. We introduce and prove the equivalence of several conditions, capturing rank one behavior of discrete subgroups of higher rank lie groups. Readings introduction to lie groups mathematics mit. Irreducible characters of semisimple lie groups ii.
Our interest, by and large, is in a special class of discrete subgroups of lie groups, viz. Raghunathan and others published discrete subgroups of lie groups find, read and cite all the research you need on researchgate. We consider the action of a semisimple lie group g on a compact manifold and more generally a borel space x, with a measure. Discrete subgroups generated by lattices in opposite. Discrete and continuous cosine transform generalized to.
Finite simple subgroups of semisimple complex lie groups. On discrete subgroups of lie groups by andrt weil received february 1, 1960 1. This solves a conjecture of margulis and extends previous works of selberg and oh. 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.
This site is like a library, use search box in the widget to get ebook that you want. 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. 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. Vanishing theorems for cohomology groups associated to discrete subgroups of semisimple lie groups. Semisimple lie groups and discrete subgroups by robert j.
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