Discrete subgroups of semisimple lie groups djvu for mac

On the first cohomology of discrete subgroups of semi. Buy discrete subgroups of lie groups and applications to moduli. The final chapter proves the basic theorems on maximal compact subgroups of lie groups. Our interest, by and large, is in a special class of discrete subgroups of lie groups, viz. Procesis masterful approach to lie groups through invariants and representations gives the reader a comprehensive treatment of the classical groups along with an extensive introduction to a wide range of topics associated with lie groups.

These are nonabelian free subgroups all of whose subgroups are either cyclic or zariski dense. Our results can be applied to the theory of algebraic groups over global fields. The rst main theorem is that the discrete subgroup sl. Finiteness results for lattices in certain lie groups greenleaf, frederick p. Basic material on affine connections and on locally or globally riemannian and hermitian symmetric spaces is covered. Discrete subgroups of lie groups and applications to moduli. Volume 7 of tata institute of fundamental research studies in mathematics tata institute of fundamental research volume 7 of studies in mathematics. The problem of classifying the real reductive groups largely reduces to classifying the simple lie groups. Dynamics in the study of discrete subgroups of lie groups.

Margulis, 9783642057212, available at book depository with free delivery worldwide. Let s be an ergodic gspace with finite invariant measure. A major achievement in the theory of discrete subgroups of semisimple lie groups is margulis superrigidity theorem. Raghunathan springerverlag berlin, new york wikipedia citation please see wikipedias template documentation for further citation fields that may be required. A detailed treatment of the geometric aspects of discrete groups was carried out by raghunathan in his book discrete subgroups of lie groups which. Once a person is a member of the parent group, they can subscribe directly to the subgroup if its permitted, they can be added directly to the.

Dimensional gap in semisimple compact lie groups via. To appear in springer lecture notes in mathematics. Examples of groups satisfying the hypotheses of theorem 17, which were not known up to now to satisfy p naive, are the burgermozes simple groups 4, which arise as lattices in products of trees. In this paper, we consider a real connected semisimple lie group g and ask whether or not a subset s of g generates g as a semigroup.

Let g be a connected semisimple lie group of finite center each of whose simple factors has rrank at least 2, or a lattice sub group of such a group. On the first cohomology of discrete subgroups of semisimple lie. Then i if a is an abelian locally compact group with no nontrivial compact subgroups, then h1s x g. Let be a zariski dense subgroup of gwhose intersection with h is an irreducible lattice in h. Mostow notes by gopal prasad no part of this book may be reproduced in any form by print, micro. A geometric construction of the discrete series for semisimple lie groups. With the goal of describing simple lie groups, we analyze semisimple complex lie algebras by their root systems to classify simple lie algebras. Suppose further that g is linear and that r contains no elements of finite order. In fact every compact semisimple lie group has at least one nonsemisimple closed subgroup.

Discrete subgroups have played a central role throughout the development of numerous mathematical disciplines. This paper introduces lie groups and their associated lie algebras. 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. 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. Shahshahani received july 7, 1971 introduction let g be a semisimple lie group acting as a group of linear transformations on the vector space v, k a maximal compact subgroup of g. A geometric construction of the discrete series for. This fact renders the centralizer of the semisimple factor in each ninvariant subgroup trivial, and effectively removes the distinction. For a large part, they summarise relevant material from knapps book 12. The group gis called reductive or semisimple if g has the corresponding property. A good basic reference is onischik and vinberg, lie groups and algebraic groups. Flows and dynamical systems on homogeneous spaces have found a wide range of applications, and of course number theory without discrete groups is unthinkable.

Discrete subgroups of semisimple lie groups ergebnisse. Readings introduction to lie groups mathematics mit. Discrete linear groups containing arithmetic groups 3 theorem 3. In section 2, we will collect some general results on lattices in locally compact groups. Of these i follows from the fact that h appears as a subrepresentation of an induced representation see the simple proof by casselman 9. Closed subgroup of a semisimple lie group mathematics. We assume a background in linear algebra, di erential manifolds, and covering spaces. Wallach, n seminar notes on the cohomology of discrete subgroups of semisimple groups. Let hbe a semisimple subgroup of a simple lie group gwith rrankh 2. A tempered distribution on g is a continuous linear functional on rg. On subsemigroups of semisimple lie groups sciencedirect. Also, the lie group r is reductive in this sense, since it can be viewed as the identity component of gl1,r. Hubsch submitted on 30 mar 2010 v1, last revised 17 jun 2017 this version, v2.

Harmonic analysis of tempered distributions on semisimple lie groups of real rank one james g. Representations on a locally convex space in this section we recall some elementary and wellknown facts about representations on locally convex spaces see 2, p. As a consequence, we get new generating results for finite simple groups of lie type and a strengthening of a theorem of borel related to the. Discrete subgroups of lie groups and applications to. Tata institute of fundamental research, bombay 1969. Its aim is to present a detailed ac count of some of the recent work on the geometric aspects of the theory of discrete subgroups of lie groups. 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. 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. Readers should be familiar with differential manifolds and the elementary theory of lie groups and lie algebras. A geometric construction of the discrete series for semisimple lie groups 3 k local integrability of the harishchandra characters.

Find materials for this course in the pages linked along the left. Subgroups have all the functionality of normal groups, with the exceptions that to be a member of a subgroup, you must be a member of the parent group, and you cannot invite people to join subgroups. Put g1 radk and g2 gg1, a connected semisimple lie group. If g is a connected lie group, then a lattice in g is a discrete subgroup. In particular, semisimple lie algebras are reductive. Finite simple subgroups of semisimple complex lie groups. Let g nakbe the iwasawa decomposition of gas above and p 0 an. A lie algebra is reductive if and only if it is the direct sum of an abelian and a semisimple lie algebra. Lattices in semisimple lie groups a theorem of wang. Discrete subgroups of semisimple lie groups gregori a. Polycyclic groups and arithmeticity of lattices in solvable lie groups.

We show that with one possible exception there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. Harmonic analysis of tempered distributions on semisimple. On zninvariant subgroups of semisimple lie groups open. 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. A similar classification holds for compact real semisimple lie groups, each of which is imbedded in a unique complex semisimple lie group as a maximal compact subgroup see. Discontinuous group actions and the study of fundamental regions are of utmost importance to modern geometry. A torus is not semisimple since it is abelian, and hence its lie algebra has nontrivial solvable ideals. Harishchandra has defined the schwartz space, vg, on g. Discrete subgroups of semisimple lie groups by gregori a. Lie groups has been an increasing area of focus and rich research since the middle of the 20th century. This can be seen as a compact analog of ados theorem on the representability of lie algebras. All lattices in a nilpotent lie group are uniform, and if is a connected simply connected nilpotent lie group equivalently it does not contain a nontrivial compact subgroup then a discrete subgroup is a lattice if and only if it is not contained in a proper connected subgroup this generalises the fact that a discrete subgroup in a vector space is a lattice if and only if it spans the vector space.

Introduction in this exposition, we consider construction and classi cation of lattices i. Rather importantly, the ninvariant regular subgroups are necessarily of maximal rank and include rank abelian factors u 1, where h i is the semisimple factor of the ninvariant regular subgroup. Then r acts without fixed points on the left on the symmetric space x gk, and can therefore be identified with the fundamental. In particular, every connected semisimple lie group meaning that its lie algebra is semisimple is reductive. Papers presented at the bombay colloquium, 1973, by baily. It is possible to develop the theory of complex semisimple lie algebras by viewing them as the complexifications of lie algebras of compact groups. We then pass to the corresponding compact lie subgroups.