Conformal geometry of discrete groups and manifolds
A geometry in the sense of Felix Klein's Erlangen program is a pair \((X,G)\) where \(X\) is a manifold and \(G\) a Lie group acting transitively on \(X\). An \((X,G)\)-manifold is a manifold \(M\) composed of open subsets of \(X\) such that the transition functions are restrictions of elements of \(G\) to open subsets of \(X\). A conformal structure on a manifold is modelled on \((S^n, \text{Möb}(n))\) where \(\text{Möb}(n)\) denotes the group of Möbius (or conformal) transformations of the \(n\)-sphere \(S^n\). Equivalently, a conformal (or better, conformally flat) structure is a conformal class of Riemannian metrics, each locally conformally equivalent to the standard flat metric. In fact, in dimensions \(n \geq 3\), by a theorem of Liouville any conformal map between open connected subsets of \(S^n\) is the restriction of a Möbius transformation. In dimension 2, by the classical uniformization theorem for Riemann surfaces, each conformal structure on a surface is globally represented by a Riemannian metric of constant curvature, i.e. modelled on one of the three classical geometries: spherical, Euclidean or hyperbolic. In any dimension these three geometries are specializations of the conformal geometry \((S^n, \text{Möb}(n))\). In dimension 3, many of the 3-manifolds belonging to one of Thurston's eight 3-dimensional (locally homogeneous Riemannian) geometries admit conformal structures (but not the solvable and nilpotent manifolds). In dimensions \(n \geq 3\), conformal structures are in general less rigid than locally homogeneous Riemannian structures; moreover, the conformal approach unifies different geometric structures admitting conformal models. The present book is a systematic study of conformal and hyperbolic structures on manifolds, in dimensions \(n \geq 3\), and of discrete groups of Möbius transformations (Kleinian groups); a unifying aspect of the subject is that of the developing map \(d: \widetilde M \to S^n\), where \(\widetilde M\) denotes the universal covering of a manifold \(M\), and of the holonomy representation \(d_*: \pi_1(M) \to \text{Möb}(n)\). Similar as various other books on related subjects (Benedetti-Petronio, Ratcliffe, Krushkal-Apanasov-Gusevskij), the book is influenced by Thurston's famous Princeton Lectures on the Geometry and Topology of 3-manifolds (part of which appeared as a book recently). It is one of the main points and differences of the present book that it shifts the more general and flexible notion of conformal structure into the center of the development, and that it considers systematically also the higher dimensional case. On 500 pages, it collects a wealth of information on the subject, in particular it may serve as a reference book and a guide to the literature of the subject. The style varies between basic definitions and elementary expositions, many examples, theorems with proofs, with sketched proofs and, in the later chapters, without proofs but with a serious effort to explain the background and the main ideas. The more general parts of the book give a rather complete picture of the present state of the art, the more specialized arguments reflect the personal taste of the author which is strongly influenced by the Russian and, in particular, the Novosibirsk school (in the preface, the author speaks of the ''golden years'' of mathematics in Novosibirsk in the seventies and eighties; in this context, one may mention also the recent book of Kapovich on Thurston's hyperbolization theorem for Haken 3-manifolds). The chapters and some of their main contents are the following: Geometric structures (conformal geometry of spheres, hyperbolic geometry, 3- and 4-dimensional geometries, geometry of orbifolds); Discontinuous groups of homeomorphisms (convergence groups, fundamental domains, limit sets); Basics of hyperbolic groups and manifolds (Margulis lemma and splittings, convex hull constructions, tesselations and reflection groups); Geometrical finiteness; Kleinian manifolds (combination theorems, ends); Uniformization (uniformizable conformal structures, conformal uniformization of Seifert manifolds and graph manifolds, sums of conformal structures); Theory of Deformations (quasi-Fuchsian structures, bendings and cone deformations). This is a useful and challenging book. The first three chapters may serve as a general introduction to the subject, the whole book as a source of information and as a reference for the area of conformal and hyperbolic geometry of manifolds and of Kleinian groups.
scientific article; zbMATH DE number 1505373