Synopses & Reviews
Non-compact Riemannian symmetric spaces and their quotients by arithmetically defined groups of isometries occur in many parts of mathematics. For many purposes it has been necessary to compactify them. This monograph attempts to give a systematic, and in part new exposition of these compactifications, of new ones, their interrelations and of the context out of which they arose. Compactifications of the more general semisimple symmetric spaces are also considered. The book is divided into three main parts. Part I is devoted to five types of compactifications, some related even isomorphic, all G- spaces, of the quotient X=3DG/K of a semisimple linear real Lie group G with finitely many connected components by a maximal compact subgroup K. The second part treats compactifications of the quotients Gamma\X, where Gamma is an arithmetic subgroup of G, assumed to be defined over the field Q of the rational numbers. In the third part, three new types of compactifications are examined: the direct constructions of T. Oshima and T. Oshima--T. Sekiguchi, the gluing of a certain number of copies of a compact manifold with corners, and the real points of the so-called wonderful compactification of the complexification of X, or more generally G/H. The compactification of noncompact Riemannian symmetric spaces leads to a rich area of research in which many mathematical disciplines come together: algebraic topology, geometry, number theory and representation theory. Familiarity with the theory of real semisimple Lie groups and symmetric spaces, and in Part II, of linear algebraic groups over Q is assumed although much material is recalled along the way. Of interest and use to researchers and graduate students in Lie Theory or Representation Theory.
Review
From the reviews: "In the book under review the authors pursue three chief goals: to give a comprehensive overview of existing compactifications ... to explain the relations among them and to provide a uniform construction. ... The style is user-friendly ... . It can be highly recommended and will be very useful to anyone, graduate student or research mathematician, interested in the geometry and topology of (locally) symmetric spaces."(Enrico Leuzinger, Mathematical Reviews, Issue 2007 d)
Synopsis
Symmetric spaces and locally symmetric spaces occur naturally in many branches of mathematics, and their intensive study by various methods has been motivated by these applications. This monograph attempts to provide a comprehensive survey and systematic exposition of these compactifications, their interrelations, the context in which they arise, and the reasons why there are so many different kinds. The authors provide uniform constructions of most known compactifications of both symmetric and locally symmetric spaces, together with some new compactifications, with an emphasis on their geometric and topological aspects.The book is divided into three main parts corresponding to different classes of compactifications. Part I covers the compactifications of Riemannian symmetric spaces and their quotients by arithmetically defined groups of isometries, Part II focuses on compact smooth manifolds, and Part III studies the compactifications of locally symmetric spaces and the relations to their metric and spectral properties. Throughout, the authors demonstrate how this rich area of research revolves around a crossroads of mathematical disciplines: algebraic topology, geometry, number theory and representation theory.Familiarity with the theory of real semisimple Lie groups and symmetric spaces is assumed, and in later parts that of linear algebraic groups over Q, though much material is recalled along the way. Both graduate students and researchers interested in Lie theory or representation theory will find the work a valuable reference.
Synopsis
Noncompact symmetric and locally symmetric spaces naturally appear in many mathematical theories, including analysis (representation theory, nonabelian harmonic analysis), number theory (automorphic forms), algebraic geometry (modulae) and algebraic topology (cohomology of discrete groups). In most applications it is necessary to form an appropriate compactification of the space. The literature dealing with such compactifications is vast. The main purpose of this book is to introduce uniform constructions of most of the known compactifications with emphasis on their geometric and topological structures. The book is divided into three parts. Part I studies compactifications of Riemannian symmetric spaces and their arithmetic quotients. Part II is a study of compact smooth manifolds. Part III studies the compactification of locally symmetric spaces. Familiarity with the theory of semisimple Lie groups is assumed, as is familiarity with algebraic groups defined over the rational numbers in later parts of the book, although most of the pertinent material is recalled as presented. Otherwise, the book is a self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to diverse fields of mathematics.
Synopsis
Introduces uniform constructions of most of the known compactifications of symmetric and locally symmetric spaces, with emphasis on their geometric and topological structures Relatively self-contained reference aimed at graduate students and research mathematicians interested in the applications of Lie theory and representation theory to analysis, number theory, algebraic geometry and algebraic topology
Table of Contents
* Preface * Introduction Part I: Compactifications of Riemannian Symmetric Spaces * Review of Classical Compactifications of Symmetric Spaces * Uniform Construction of Compactifications of Symmetric Spaces * Properties of Compactifications of Symmetric Spaces Part II: Smooth Compactifications of Semisimple Symmetric Spaces * Smooth Compactifications of Riemannian Symmetric Spaces G / K * Semisimple Symmetric Spaces G / H * The Real Points of Complex Symmetric Spaces Defined Over R * The DeConcini-Procesi Compactification of a Complex Symmetric Space and its Real Points * The Oshima-Sekiguchi Compactification of G / K and Comparison with G/Hw (R) Part III: Compactifications of Locally Symmetric Spaces * Classical Compactifications of Locally Symmetric Spaces * Uniform Construction of Compactifications of Locally Symmetric Spaces * Properties of Compactifications of Locally Symmetric Spaces * Subgroup Compactifications of o \ G * Metric Properties of Compactifications of Locally Symmetric Spaces o \ X * References * Index