05454 - Discrete Subgroups of Lie Groups [Raghunathan].pdf

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M. S. Raghunathan
Discrete Subgroups
of Lie Groups
Springer-Verlag Berlin Heidelberg
New
York
1972
M. S. Raghunathan
Tata Institute of Fundamental Research
Bombay, India
AMS Subject Classifications (1970):
Primary
22 E 40
Secondary 32 N XX, 20 H XX
ISBN 3-540-05749-8 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-05749-8 Springer-Verlag New York Heidelberg Berlin
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C
by Springer-Verlas Ber-
Un Hcidelbers 1972. Library of Consress Catalos Card Number 71·189389. Printed in Germany,
TypesettlnS, prlntlns and blndlns: Unlvenltltsdruckeral H. Stortz AG, WOrzburg.
To
my
parents
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Preface
This book originated from a course of lectures given at Yale University
during 1968-69 and a more elaborate one, the next year, at the Tata
Institute of Fundamental Research. 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. Our interest, by and large, is in a
special class of discrete subgroups of Lie groups, viz., lattices (by a
lattice in a locally compact group G, we mean a discrete subgroup
H
such that the homogeneous space
GIH
carries a finite G-invariant
measure).
It is assumed that the reader has considerable familiarity with Lie
groups and algebraic groups. However most of the results used frequently
in the book are summarised in "Preliminaries"; this chapter, it is hoped,
will be useful as a reference.
We now briefly outline the contents of the book. Chapter I deals with
results of a general nature on lattices in locally compact groups. The
second chapter is an account of the fairly complete study of lattices in
nilpotent Lie groups carried out by Malcev. Chapters III and IV are
devoted to lattices in solvable Lie groups; most of the theorems here are
due to Mostow. In Chapter V we prove a density theorem due to Borel:
this is the first important result on lattices in semisimple Lie groups.
The next two chapters are somewhat of a digression from the main trend
of the book and may be omitted without any break in continuity (though
occasionally one of the theorems proved in Chapter VI is used elsewhere).
Chapter VI contains some general theorems on finitely generated sub-
groups of Lie groups while Chapter VII is on cohomological results for
discrete subgroups of Lie groups (work of Mostow on the cohomology of
solv-manifolds and some results of Weil and Matsushima-Murakami on
compact locally symmetric spaces). Chapter VIII plays a central role; it is
indispensable for the rest of the book. The main result here is a theorem
due to Zassenhaus which has proved to be basic for the entire theory.
Among other results proved here are theorems due to H. C. Wang and
L. Auslander which show, among other things, that, at least up to a point,
the study of lattices in general Lie groups can be split into studying those

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