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Large scale geometry. (English) Zbl 1264.53051

EMS Textbooks in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-112-5/hbk). xiv, 189 p. (2012).
Large scale geometry is essentially the study of geometric objects viewed from afar. As the authors explain in the preface “two objects are considered to be the same if they look roughly the same from a large distance”. This is as far as we know the first book that aims to give the reader a full picture on this subject although there is a paper by the second author that is actually a survey on the subject [G. Yu, Contemp. Math. 546, 305–315 (2011; Zbl 1245.53043)].
Large scale geometry ideas originated in the works of several mathematicians: [G. D. Mostow, Ann. Math. Stud. No. 78. (1973; Zbl 0265.53039); Publ. Math., Inst. Hautes Étud. Sci. 34, 53–104 (1968; Zbl 0189.09402)], G. A. Margulis [Sov. Math., Dokl. 11, 722–723 (1970); translation from Dokl. Akad. Nauk SSSR 192, 736–737 (1970; Zbl 0213.48202); Probl. Peredaci Inform. 9, No. 4, 71–80 (1973; Zbl 0312.22011); Monatsh. Math. 90, 233–236 (1980; Zbl 0425.43001)] and A. S. Švarc [Dokl. Akad. Nauk SSSR 105, 32-34 (1955; Zbl 0066.15903)] related to Mostow’s rigidity theorem and its generalizations and also J. W. Milnor [J. Differ. Geom. 2, 1–7 (1968; Zbl 0162.25401)] and J. A. Wolf [J. Differ. Geom. 2, 421–446 (1968; Zbl 0207.51803)] related to growth of groups.
The main goal of this book is to give an introduction as complete as possible on the subject and to show how large scale geometry methods can be used in index theory, more precisely in the extension of Atiyah-Singer index theorem to non-compact manifolds. It is organized in eight chapters. The first chapter provides an introduction to the basic notions needed including a section on Gromov hyperbolicity.
The second chapter explains the notions of asymptotic dimension and decomposition complexity.
The third chapter introduces the notion of amenability. This notion allows one to average over infinite groups. This chapter includes the geometric definition by Føner, the functional definition by Hulanicki and Reiter and the definition involving an invariant mean.
Chapter 4 introducing the property A. This property can be considered a weak version of amenability. Besides the definition and basic properties there is a section on the Higson-Roe condition. In this section it is shown that amenability implies property A. In the following section it is shown that finite asymptotic dimension implies property A. There are many examples of spaces with property A included within the previous sections of the chapter. The final sections of this chapter gives the example of two groups that do not have property A; one of these examples uses residually finite groups, the other finite groups.
Chapter 5 deals with coarse embeddings into Hilbert spaces. Since a metric space that has property A embeds coarsely into the Hilbert space it is natural to look for examples of embeddable spaces without property A. The authors give examples which are locally finite and examples with bounded geometry (section three). In Section 4 the authors introduce convexity and reflexivity. The next section deals with the proof of the converse of the following statement: If a bounded geometry metric space \(X\) embeds coarsely into a Hilbert space with bounds \(\rho _-\) and \(\rho _+\), then the restriction of the embedding to any subset \(A \subseteq X\) is also a coarse embedding with the same bounds. Section 6 deals with expanders (finite graphs which are highly connected but sparse at the same time). These sets will serve as examples of spaces which do not embed coarsely into a Hilbert spaces. In order to look for more examples, the authors give a geometric characterization of non-embeddability in Section 7. To measure how the spaces embed in an Hilbert space the notion of compression number is introduced in Section 8 and the last section explains how compression greater then \(1/2\) implies property A.
In Chapter 6 the authors discuss affine isometric group actions on Banach spaces. The chapter includes a short introduction to affine isometric actions, Gromov’s a-T-menability, (also known as the Haagerup property), Kazdan’s property (T) and its relation with fixed points. This chapter includes the construction of more examples of expanders. Proving that these examples satisfy property (T) is, quoting the authors, “highly non-trivial” and in the last section a sufficient condition is given, relating property (T) with spectral conditions; more precisely relating it to the eigenvalues of the Laplacian on a certain finite graph.
In Chapter 7 the authors explore various versions of coarse homology groups. There are sections on coarse locally-finite homology and on uniformly finite homology. Since the vanishing of the fundamental class can be described in terms of Eilenberg swindles, there is a section on the subject which also includes Ponzi schemes. The authors include an application of uniformly finite homology in the construction of aperiodic tiles and relate it with non-amenable spaces. The last sections of this chapter deal with coarsening homology theories in a more detailed point of view. It is written for readers with knowledge on algebraic topology.
The last chapter is a survey of applications on topological rigidity, geometric rigidity and index theory.
At the end of each chapter, the authors includes some notes and remarks that give further information on the subjects dealt with, as well as some exercises and sometimes even open questions.

MSC:

53C24 Rigidity results
51-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry
51F99 Metric geometry
20F69 Asymptotic properties of groups
19K56 Index theory
57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
46L87 Noncommutative differential geometry
58B34 Noncommutative geometry (à la Connes)
58J20 Index theory and related fixed-point theorems on manifolds
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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