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Geometric group theory. An introduction. (English) Zbl 1426.20001
Universitext. Cham: Springer (ISBN 978-3-319-72253-5/pbk; 978-3-319-72254-2/ebook). xi, 389 p. (2017).
The book consists from 9 chapters in three parts and an appendix (in Part IV).
Part I (Chapter 2) begins with the definition of groups, in few pages running the basic notions to end up to its aim, the definition of a free group and the resulting notion of the presentation of a group with generators and relations. The chapter concludes with a first development of amalgamated free products and HNN-extensions.
A fundamental question of geometric group theory is how groups can be viewed as geometric objects. Part II (Chapters 3, 4 and 5) is devoted to study this question. In Chapter 3, the notion of the Cayley graph of a (finitely generated) group is developed and, apart the definitions and the properties of Cayley graphs, the chapter focuses to the tree of a free group.
In Chapter 4, the action of a group on graphs is developed and, more specially, the (natural) action of a group on its Cayley graph is studied. This leads to the characterization of a free group by its (free) action on a tree. Applications of this characterization are the Nielsen-Schreier theorem and the corollaries of this theorem. Another (sufficient) criterion (the ping-pong lemma) for the freeness of a group is proved and as an application it is proved that linear groups contain free groups as subgroups. Finally, a sketch of a proof of the Tits alternative is given.
If \(S\) is a generating set of a group \(G\), then the associated Cayley graph \(\mathrm{Cay}(G,\,S)\) is endowed with a metric (the word metric with respect to the generating set \(S\)). This metric depends on the set \(S\). So, since the group would be considered as geometric object independent of the chosen generating set, this difficulty must be overcome. This is obtained by passing to large scale geometry. Chapter 5 is devoted to this purpose. The notions of isometry, bilipschitz equivalence and quasi-isometry of metric spaces are developed and are applied to characterize groups up to bilipschitz equivalence and quasi-isometry. Here, the nodal point is the Švarc-Milnor lemma, which says that a ‘nice’ action of a group \(G\) on a ‘nice’ metric space \((X,\,d)\) ensures that the group \(G\) is finitely generated by a set \(S\) and that the spaces \((G,\,d_{S})\) and \((X,\,d)\) are quasi-isometric. The topological version of this lemma is:
Let \(G\) be a group acting by isometries on a (non-empty) proper and deodesic metric space \((X,\, d)\). Furthermore, suppose that this action is proper and cocompact. Then \(G\) is finitely generated, and for all \(x\in X\) the map \(G\,\longrightarrow\, X\) with \(g\,\longrightarrow\, g\cdot x\) is a quasi-isometry.
This version is used in many applications and, since there is not an analogous for the bilipschitz equivalence, the quasi-isometry is more appropriate in the study of finitely generated groups as geometric objects.
The chapter concludes with the main problem in geometric group theory, the classification of finitely generated groups up to quasi-isometry. So the notion of the quasi-isometry invariants is introduced and a first task of the geometric properties of groups is given.
In Part III, the main quasi-isometry invariants are studied. In Chapter 6, the growth type of a group is discussed. The growth function of a finitely generated group \(G\) with respect to a finite generating set \(S\) is a function which “measures” the number of elements that are contained in a sphere with center the trivial element and radius \(r\). Although a growth function is depended upon the chosen generating set, it is proved that quasi-isometric groups have quasi-equivalent growth functions. This yields to the growth type of a group.
The growth type of a group is the (common) quasi-equivalence class of all growth functions of the group with respect to its finite generating sets. The growth type of a group is a geometric property and there are three growth types of groups. The exponential growth, the polynomial growth and the intermediate growth. The growth type of a finitely generated free group is exponential. Therefore the growth type of any finitely generated group does not exceed the exponential type.
There exist intermediate growth-type groups, for example the first Grigorchuk group.
A characterization for the finitely generated groups of polynomial growth type is given by Gromov.
Finitely generated groups have polynomial growth if and only if they are virtually nilpotent.
One of the most important results in geometric group theory.
Chapter 7 begins with a report on the curvature of (geodesic) spaces and focuses to hyperbolic spaces and particularly on the Cayley graphs of groups. It is proved that the curvature of the Cayley graph of a group does not depend on the choosen generating set. This guides to the development of the theory of negatively curved finitely generated (hyperbolic) groups. More generally, it is proved that the negative curvature of groups (considered as geometric objects) is invariant under quasi-isometries.
The hyperbolicity of groups reflects to algebraic properties of groups. For example, hyperbolic groups have solvable word problem. The centralizer of an infinite order element of a hyperbolic group is small.
The chapter concludes with a brief outlook on non-positively curved (CAT(0)) groups and pointed out that CAT(0) is not a quasi-isometry invariant among finitely generated groups.
The geometry at infinity is involved in the study of groups. As it is known to a geodesic (more generally to a quasi geodesic) metric space \(X\) it is assigned the space \(\mathrm{Ends}(X)\) (\(\mathrm{Ends}_{Q}(X)\)) of (quasi) ends of \(X\). This space is endowed by a topology. In Chapter 8, this is developed briefly but understandably giving the point that quasi-isometric embebddings (quasi-isometries) between quasi-geodesic spaces \(X\) and \(Y\) induce continuous maps (homeomorphisms) between the relevant spaces of ends \(\mathrm{Ends}(X)\) and \(\mathrm{Ends}(Y)\). These results are applied to finitely generated groups by studying their Cayley graphs. The results are impressive.
Let \(G\) be a finitely generated group. Then \(G\) has 0, 1, 2 or infinitely many ends. More exactly there are characterizations.
1. A finitely generated group has no ends if and only if it is finite.
2. A finitely generated group has two ends if and only if it is virtually \(\mathbb{Z}\).
3. Stallings’ decomposition theorem. A finitely generated group has infinitely many ends if and only if it splits over a finite group.
An analogous (but finer) construction is the Gromov boundary. Here, this notion is introduced firstly for any quasi-geodesic metric space. But in the case of proper hyperbolic metric spaces where the boundary can be expressed in terms of geodesic rays instead of quasi-geodesic rays, the results are more tangible and can be applied to Cayley graphs of hyperbolic groups and it is proved (a Tits alternative for hyperbolic groups) that:
If \(g,\,h\) are two elements of infinite order of an hyperbolic group \(G\), then the subgroup \(\langle\,g,\,h\,\rangle\) generated by them is either virtually \(\mathbb{Z}\) or there are \(m,\,n\in \mathbb{N}\) such that \(\langle\,g^{m},\,h^{n}\,\rangle\) is free of rank 2. In conclusion, an hyperbolic group is either virtually cyclic or contains a free group of rank 2.
The chapter concludes with a brief report on applications of boundary methods to acylindrically hyperbolic groups and to Mostow rigidity and ends with a reference to the Borel conjecture: “Closed connected manifolds with contractible universal covering space are homotopy equivalent if and only if they are homeomorphic”.
The amenability of groups is studied in the last chapter (Chapter 9 ) of the book. The first definition of amenable groups is given via invariant means. This definition is enough to give some characteristic examples of amenable and non-amenable groups as well to prove some properties concerning amenable groups. Here, the (relative) von Neumann problem is referred: “Does every non-amenable group contain a free group of rank 2?”. This problem was answered negatively by Ol’shanskii. But has positive answer in many interesting cases.
Equivalent characterizations of amenability related to geometric properties of groups are given.
Let \(G\) be a finitely generated group \(G\) with \(S\) a finite generating set. Roughly speaking, a subset \(F\) of \(G\) is Folner if its boundary (considered as a subset of the Cayley graph \(\mathrm{Cay}(G,\,S)\)) is “small” relatively to the “volume” of \(F\).
A finitely generated group \(G\) with \(S\) a finite generating set is amenable if and only if the metric space \(\mathrm{Cay}(G,\,S)\) admits a sequence of non empty finite Folner sets.
In general, a UDBG (see Definition 5.6.11) metric space \((X,\,d)\) is called amenable if it admits a sequence of non empty finite Folner sets.
Another geometric characterization of amenability is based on decomposition paradoxa. Loosely, a paradoxical decomposition of a group is a decomposition into finitely many disjoint subsets such that these subsets can be rearranged by translations into two copies of the group.
A group is paradoxical (namely admits a paradoxical decomposition) if and only if it is amenable (Tarski’s theorem).
Let \(G\) be a group which acts on a set \(X\). Similarly, a paradoxical decomposition of the set \(X\) is a decomposition into disjoint subsets such that these subsets can be rearranged using the group action into two copies of the set. Consider the (natural) action of the special orthogonal group \(\mathrm{SO}(3)\) on the sphere \(S^{2}\).
Banach-Tarski theorem for the sphere: The sphere \(S^{2}\) admits a paradoxical decomposition with respect the canonical action of \(\mathrm{SO}(3)\) on \(S^{2}\).
Characterizations of amenability in terms of uniformly finite homology and bounded cohomology are also given. Amenability is a geometric quasi-isometry invariant property. Two quasi-isometric UDBG spaces (finitely generated groups) must be both amenable ( or not-amenable).
The chapter concludes with a discussion on the relation between quasi-isometry and bilipschitz equivalence.
As it is pointed out, there are groups which are quasi-isometric but not bilipschitz equivalent. However, for non-amenable groups there is a rigidity phenomenon. Every quasi-isometry between non-amenable UDBG spaces (or finitely generated non-amenable groups) is at finite distance from a bilipschitz equivalence.
In the appendix, a reference to three fundamental notions in mathematics is given. The fundamental group of a space, the group (co)homology and the hyperbolic plane. There, these notions are developed in a very short but concise way. Here, the reader can trace back to refresh these notions and to have an incentive for a further study.
It is not possible to develop all proofs thoroughly and in detail. Therefore in many cases an outline of the proof is given referring to original sources. The rich but selected bibliography at the end is valuable for the reader and constitutes a complement of the book.
At the end of each chapter, many targeted exercises are available. The purpose of these exercises is not only to examine the reader what he has learned by reading the book, but also to give him the motivation for further studying. Besides, many of them (as pointed out by the author) come as a supplement to the text and are relevant to ongoing research topics. The structure of the chapters can make the reader independent, thus the book can be used “outside of the classroom” for self-teaching by both young researchers and experienced scholars. The book is well written, and along with [C. Druţu and M. Kapovich, Geometric group theory. With an appendix by Bogdan Nica. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1447.20001)] it is ready to fill a gap in the literature for such an interesting and active branch of mathematics.

20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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