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Compactifications of symmetric and locally symmetric spaces. (English) Zbl 1100.22001
Mathematics: Theory & Applications. Basel: Birkhäuser (ISBN 0-8176-3247-6/hbk). xiii, 479 p. (2006).
The book under review is mainly concerned with symmetric spaces of non-compact type and their quotients by arithmetic transformation groups, the latter of which are often referred to as locally symmetric spaces. These classes of topological spaces naturally (and abundantly) appear in many areas of modern mathematics, be it as important special manifolds or as classifying spaces of particular geometric structures. Prominent examples of their ubiquity can be found in non-abelian harmonic analysis, in the representation theory of Lie groups, in the arithmetic theory of automorphic forms, in the algebro-geometric theory of moduli spaces, and in the cohomology theory of discrete groups. In most concrete applications, it is indispensable to deal with suitable compactifications of such non-compact spaces, mainly for computational reasons in specific contexts. Accordingly, and due to the huge variety of problems connected with non-compact symmetric and locally symmetric spaces, the literature related to various possible compactifications of such spaces is tremendously vast and polymorphic. In fact, over the past fifty years, it has grown into a veritable avalanche of research reports in different fields of mathematics, with the names of many leading mathematicians being involved. Certainly, this high degree of diversity in the compactification theory of symmetric and locally symmetric spaces, by this time, virtually cried for appropriate systematization and uniformization, at least to some extent, and the present monograph was written in order to serve that purpose. As it is pointed out in the preface, the basic plan of this book was worked out by the two authors in 2002, based on a series of foregoing joint research papers on the subject [cf. A. Borel and L. Ji, Progr. Math. 229, 1–67 (2005; Zbl 1088.53034)]. One main part of the book (Part II) was mostly written by the first author, the unforgettable Armand Borel (1923–2003), before his sudden death on August 11, 2003, whereas the task of finishing their common work, thereby transmitting the legacy of one of the great mathematicians of the 20th century, fell to the second author. The purposes of this book are outlined as follows:
1. To explain how various compactifications were constructed, why they have particular properties and structures of their boundary, and to what extent different types of compactifications are related to each other.
2. To provide uniform constructions of almost all known types of compactifications, on the one hand, and to exhibit new compactifications, on the other hand.
3. To re-examine geometrical properties of compactifications from a group-theoretic viewpoint, with an outlook to concrete (further) applications of them.
This just as rewarding as pioneering program is carried out in a masterly, very systematic and detailed manner, emphasizing at least five unifying major themes as strategic-methodological fundamentals. There are basically three different types of compactifications, and most of the classical compactifications belong to one of them. More precisely, the authors consider (i) compact spaces containing a symmetric space as an open dense subset, (ii) compact smooth analytic manifolds containing a disjoint union of at least two but finitely many symmetric spaces as an open dense subset, and (iii) compact spaces that contain a locally symmetric space as an open dense subset. According to these three types of compactifications, the book is divided into three parts devoted to each of them, respectively.
Part I deals with compactifications of Riemannian symmetric spaces. In the three chapters constituting this part, the authors review most of the known compactifications of this type, give then a uniform construction of them, exhibit – via this approach – several new compactifications, and finally study the relations between all these compactifications and their structural properties. This includes a detailed discussion of the geodesic compactification and Tits buildings, the Karpelevič compactification, the Baily-Borel compactification, the Satake compactifications, the Furstenberg compactifications, the Martin compactifications, the real Borel-Serre partial compactification, the dual-cell compactification, and several new constructions of the maximal Satake compactification.
Part II is mainly devoted to smooth compactifications of semi-simple symmetric spaces. Here the symmetric spaces under consideration are not necessarily Riemannian, their compactifications appear as closed smooth manifolds, but the symmetric spaces themselves are possibly not dense therein. This part encompasses Chapters 4 to 8. The authors first provide a general method of self-gluing a manifold with corners into a closed manifold, which is then used to reconstruct the classical Oshima compactification of a Riemannian symmetric space from the maximal Satake compactification. After a thorough study of basic properties of semi-simple symmetric spaces, real loci of complex symmetric spaces are described in terms of Galois cohomology. This is used, in the sequel, to determine the real locus of the so-called “wonderful compactification” of a complex symmetric space à la C. De Concini and C. Procesi [Lect. Notes Math. 996, 1–44 (1983; Zbl 0581.14041)]. Combined with the self-gluing method developed before, this leads to a fine analysis of the real-analytic structure on the Oshima compactification, and the results of this chapter also clarify why the study of compactifications of Riemannian symmetric spaces naturally leads to compactifications of non-Riemannian semi-simple symmetric spaces. In the concluding section of Part II, the classical Oshima-Sekiguchi compactification [T. Oshima and J. Sekiguchi, Adv. Stud. Pure Math. 4, 433–497 (1984; Zbl 0577.17004)] is related to the real locus of the De Concini-Procesi wonderful compactification by a finite-to-one map.
Part III turns to compactifications of locally symmetric spaces, which are thoroughly treated in Chapters 9 to 13. At the beginning, the most important classical compactifications of this type are recalled and discussed, namely the Satake compactification(s), the Baily-Borel compactification, the Borel-Serre compactification, the reductive Borel-Serre compactification, and the common toroidal compactifications of locally symmetric spaces. According to the prevailing arithmetic-geometric nature of this topic, the necessary facts on rational parabolic subgroups, arithmetic subgroups, and their related reduction theories are placed in front of the exposition. Subsequently, a systematic, uniform approach to compactifying locally symmetric spaces is presented. This is then applied to reconstruct the classical compactifications described before, within the unified framework. By the way, the authors’ uniform approach, in this context, is very similar to the uniform method of compactifying Riemannian symmetric spaces (as presented in Part I), thereby demonstrating the beautiful coherence of the entire theory. The following chapter treats further properties of compactified locally symmetric spaces, emphasizing close relations between different constructions. In particular, it is shown how self-gluing of the Borel-Serre compactification yields the so-called Borel-Serre-Oshima compactification of a locally symmetric space. Other types of compactifications for homogeneous spaces $$\Gamma\setminus X$$ are studied, too, namely the so-called subgroup compactifications.
The final chapter of the book deals with metric properties of compactified locally symmetric spaces. This important topic is not only motivated from the viewpoint of extremal properties in Riemannian geometry, but also in regard of its crucial role in algebraic geometry (with respect to extensions of period mappings for degenerating families of algebraic varieties) and in the spectral theory of automorphic forms. The discussion in this chapter culminates in concrete applications to a conjecture of C. L. Siegel on metrics on Siegel sets [cf. C. L. Siegel, Zur Reduktionstheorie quadratischer Formen, Publ. Math. Soc. Japan, vol. 5 (1959; Zbl 0097.00901); also in: Gesammelte Werke III, pp. 274–327], for which a confirming proof is given, as well as in new results on hyperbolic compactifications and extensions of holomorphic maps from the punctured disc to Hermitian locally symmetric spaces.
Apart from a very enlightening and motivating introduction to the whole book, enriched by numerous historical remarks, each part comes with its own individual introduction, and the bibliographical references and hints are more than plentiful.
However, as one can see from the very contents, and as the authors duly point out in the general introduction to their book, there are some related topics that could not be covered. Nevertheless, those topics are carefully listed, and a classification of references for a deeper study of them is provided. In general, the present book focuses on the geometric aspects of compactification theory, and it does so in a nearly encyclopedic manner. Although some familiarity with the theory of Lie groups and algebraic groups is assumed, the present book may be seen as a largely self-contained reference for researchers and seasoned graduate students in the field. As for a suitable introduction to this monograph, the reader is advised to consult the related collection of lecture notes “Lie groups and automorphic forms”, edited by Lizhen Ji, Jian-Shu Li, and Shing-Tung Yau [AMS/IP Studies in Advanced Mathematics, v. 37, AMS, Providence, RI, 2006, ISBN 978-0-8218-4198-3], which contains further survey lectures by A. Borel and by L. Ji, and which has just been published. Altogether, it can simply be stated that the book under review is the finally existing, comprehensive and unifying standard text on compactifications of symmetric and locally symmetric spaces, and an utmost valuable guide to the entire field of research likewise.

##### MSC:
 22-02 Research exposition (monographs, survey articles) pertaining to topological groups 22F30 Homogeneous spaces 11Fxx Discontinuous groups and automorphic forms 22E40 Discrete subgroups of Lie groups 53Cxx Global differential geometry 32Mxx Complex spaces with a group of automorphisms 53C35 Differential geometry of symmetric spaces