Elements of asymptotic geometry.

*(English)*Zbl 1125.53036
EMS Monographs in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-036-4/hbk). xii, 200 p. (2007).

In many geometric theories (for example in Riemannian geometry) the local geometry plays an essential role. Asymptotic geometry means, roughly speaking, the study of metric spaces from a large scale point of view, where the local geometry does not come into play. For Gromov hyperbolic spaces (in particular for Gromov hyperbolic groups) it turns out that the asymptotic geometry is almost completely encoded in the boundary at infinity. The basic example of such a space is the classical hyperbolic space \(H^{n}\). A main feature of \(H^{n}\) is the deep relation between the geometry of \(H^{n}\) and the Möbius geometry of its boundary \(\partial_{\infty}H^{n}\). For example the isometries of \(H^{n}\) are in one-to-one correspondence with the Möbius transformations of \(\partial_{\infty}H^{n}\). The general hyperbolic space (in the sense of Gromov) theory is very similar to the classical hyperbolic space theory.

Concerning the book under review, only a few elements of asymptotic geometry are discussed. In the first eight chapters, the authors provide a systematic account of the basic Gromov hyperbolic space theory. Chapter 1 deals with basic notions related to geodesic metric spaces. The property of a geodesic metric space to be hyperbolic is defined in terms of triangles and the Gromov product. One proves that geodesics in hyperbolic geodesic spaces are stable.

Chapter 2 starts with a discussion of further properties of the Gromov product with the aim of deriving the \(\delta\)-inequality (see Proposition 2.1.2) for hyperbolic geodesic spaces, which allows the extension of the notion of hyperbolicity to metric spaces that are not necessarily geodesic. An important point of this discussion is the Tetrahedron Lemma, that has various applications throughout the book. Various structures attached to the boundary at infinity for any hyperbolic space are discussed: Gromov product, quasi-metrics, visual metrics and topology. This chapter contains also the proof of the local self-similarity of the boundary at infinity of every cobounded, hyperbolic, proper, geodesic space with respect to any visual metric.

In Chapter 3, the authors develop appropriate tools if one tries to choose a basepoint at infinity, the most important of which are Busemann functions. Properties of Gromov products and visual metrics based at infinity are introduced and studied. Chapter 4 treats a natural class of morphisms between hyperbolic spaces, namely, the power quasi (PQ)-isometric maps. A PQ-isometric map \(f\) between any metric spaces is quasi-isometric, and in the case of arbitrary hyperbolic spaces it is shown that \(f\) naturally induces a map between their boundaries at infinity, which is automatically quasi-Möbius (see Proposition 5.2.10).

Chapter 5 is concerned with the generalization of the classical result that every isometry of \(H^{n}\) induces a Möbius map of \(\partial_{\infty}H^{n}\) to hyperbolic geodesic spaces. Chapter 6 discusses a special kind of hyperbolic cones (a hyperbolic cone is a hyperbolic space with prescribed boundary at infinity) called hyperbolic approximations. Chapter 7 contains the proofs of three extension results, each saying that given a map with certain properties between the boundaries at infinity of hyperbolic spaces, there is a map in an appropriate class between the spaces themselves which induces the given map of the boundaries. Chapter 8 covers the proofs of the Assouad and Bonk-Schramm embedding theorems.

In the second part of the book, the authors consider various dimension type asymptotic invariants of arbitrary metric spaces and give several interesting applications, in particular, for embedding and non-embedding results concerning hyperbolic spaces. Chapter 9 treats basics of dimension theory in metric space setting: The authors discuss a number of dimensions all of which are close relatives of the classical topological or covering dimension. Chapter 10 is an exposition on estimates from below and from above, including optimal ones, for the asymptotic dimension of different classes of metric spaces. Basic properties of the linearly controlled metric dimension, which are established in Chapter 11, are used in Chapter 12, where a general and sharp embedding result of a visual Gromov hyperbolic space into a product of metric trees as well as some of its consequences are proven. The covering definition of the hyperbolic dimension of a metric space, a quasi-isometric invariant, is defined in Chapter 13, where the authors give a proof based on Sperner’s lemma that the hyperbolic dimension of any Hadamard manifold of dimension \(n\), with sectional curvature \(\leq-1\), is at least \(n\). This estimate from below allows to derive nonexistence results concerning quasi-isometric embeddings. In Chapter 14 other quasi-isometric invariants are considered, namely, the hyperbolic rank and subexponential corank, which are used for the proof of other nonembedding results.

As mentioned, the book develops basic Gromov hyperbolic space theory and emphasizes several aspects of the asymptotic geometry of arbitrary metric spaces. It is carefully written to be accessible to graduate students. The exercises are well chosen and each chapter ends with a separate section containing supplementary results related to recent developments in the subject as well as bibliographical notes. An Appendix giving a description of various models of \(H^{n}\) and explaining the classical result that there is one-to-one correspondence between the isometries of \(H^{n}\) and the Möbius transformations of the unit sphere \(S^{n-1}\). The recently published book [J. Roe, Lectures on coarse geometry. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1042.53027)] about the same topic can be considered as a complement of this book. In view of the new ideas and the activity in this area, the book is a timely one and the authors are to be complimented for bringing together major threads of the subject in a very readable account.

Concerning the book under review, only a few elements of asymptotic geometry are discussed. In the first eight chapters, the authors provide a systematic account of the basic Gromov hyperbolic space theory. Chapter 1 deals with basic notions related to geodesic metric spaces. The property of a geodesic metric space to be hyperbolic is defined in terms of triangles and the Gromov product. One proves that geodesics in hyperbolic geodesic spaces are stable.

Chapter 2 starts with a discussion of further properties of the Gromov product with the aim of deriving the \(\delta\)-inequality (see Proposition 2.1.2) for hyperbolic geodesic spaces, which allows the extension of the notion of hyperbolicity to metric spaces that are not necessarily geodesic. An important point of this discussion is the Tetrahedron Lemma, that has various applications throughout the book. Various structures attached to the boundary at infinity for any hyperbolic space are discussed: Gromov product, quasi-metrics, visual metrics and topology. This chapter contains also the proof of the local self-similarity of the boundary at infinity of every cobounded, hyperbolic, proper, geodesic space with respect to any visual metric.

In Chapter 3, the authors develop appropriate tools if one tries to choose a basepoint at infinity, the most important of which are Busemann functions. Properties of Gromov products and visual metrics based at infinity are introduced and studied. Chapter 4 treats a natural class of morphisms between hyperbolic spaces, namely, the power quasi (PQ)-isometric maps. A PQ-isometric map \(f\) between any metric spaces is quasi-isometric, and in the case of arbitrary hyperbolic spaces it is shown that \(f\) naturally induces a map between their boundaries at infinity, which is automatically quasi-Möbius (see Proposition 5.2.10).

Chapter 5 is concerned with the generalization of the classical result that every isometry of \(H^{n}\) induces a Möbius map of \(\partial_{\infty}H^{n}\) to hyperbolic geodesic spaces. Chapter 6 discusses a special kind of hyperbolic cones (a hyperbolic cone is a hyperbolic space with prescribed boundary at infinity) called hyperbolic approximations. Chapter 7 contains the proofs of three extension results, each saying that given a map with certain properties between the boundaries at infinity of hyperbolic spaces, there is a map in an appropriate class between the spaces themselves which induces the given map of the boundaries. Chapter 8 covers the proofs of the Assouad and Bonk-Schramm embedding theorems.

In the second part of the book, the authors consider various dimension type asymptotic invariants of arbitrary metric spaces and give several interesting applications, in particular, for embedding and non-embedding results concerning hyperbolic spaces. Chapter 9 treats basics of dimension theory in metric space setting: The authors discuss a number of dimensions all of which are close relatives of the classical topological or covering dimension. Chapter 10 is an exposition on estimates from below and from above, including optimal ones, for the asymptotic dimension of different classes of metric spaces. Basic properties of the linearly controlled metric dimension, which are established in Chapter 11, are used in Chapter 12, where a general and sharp embedding result of a visual Gromov hyperbolic space into a product of metric trees as well as some of its consequences are proven. The covering definition of the hyperbolic dimension of a metric space, a quasi-isometric invariant, is defined in Chapter 13, where the authors give a proof based on Sperner’s lemma that the hyperbolic dimension of any Hadamard manifold of dimension \(n\), with sectional curvature \(\leq-1\), is at least \(n\). This estimate from below allows to derive nonexistence results concerning quasi-isometric embeddings. In Chapter 14 other quasi-isometric invariants are considered, namely, the hyperbolic rank and subexponential corank, which are used for the proof of other nonembedding results.

As mentioned, the book develops basic Gromov hyperbolic space theory and emphasizes several aspects of the asymptotic geometry of arbitrary metric spaces. It is carefully written to be accessible to graduate students. The exercises are well chosen and each chapter ends with a separate section containing supplementary results related to recent developments in the subject as well as bibliographical notes. An Appendix giving a description of various models of \(H^{n}\) and explaining the classical result that there is one-to-one correspondence between the isometries of \(H^{n}\) and the Möbius transformations of the unit sphere \(S^{n-1}\). The recently published book [J. Roe, Lectures on coarse geometry. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1042.53027)] about the same topic can be considered as a complement of this book. In view of the new ideas and the activity in this area, the book is a timely one and the authors are to be complimented for bringing together major threads of the subject in a very readable account.

Reviewer: Mircea Craioveanu (Timişoara)

##### MSC:

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

20F67 | Hyperbolic groups and nonpositively curved groups |

20F69 | Asymptotic properties of groups |

54F45 | Dimension theory in general topology |

51F99 | Metric geometry |

55M10 | Dimension theory in algebraic topology |