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The authors introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the latter one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, Out(F_n), and the Cremona group. Other examples can be found among groups acting geometrically on CAT(0) spaces, fundamental groups of graphs of groups, etc. The authors obtain a number of general results about rotating families and hyperbolically embedded subgroups; although their technique applies to a wide class of groups, it is capable of producing new results even for well-studied particular classes. For instance, the authors solve two open problems about mapping class groups, and obtain some results which are new even for relatively hyperbolic groups.

Hyperbolic spaces. --- Hyperbolic groups.

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Hyperbolic spaces. --- Hyperbolic groups. --- Espaces hyperboliques --- Groupes hyperboliques --- Hyperbolic spaces --- Hyperbolic groups --- Espaces hyperboliques. --- Groupes hyperboliques.

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Chaotic behavior in systems --- Differentiable dynamical systems --- Hyperbolic spaces

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Over the past three decades there has been a total revolution in the classic branch of mathematics called 3-dimensional topology, namely the discovery that most solid 3-dimensional shapes are hyperbolic 3-manifolds. This book introduces and explains hyperbolic geometry and hyperbolic 3- and 2-dimensional manifolds in the first two chapters and then goes on to develop the subject. The author discusses the profound discoveries of the astonishing features of these 3-manifolds, helping the reader to understand them without going into long, detailed formal proofs. The book is heavily illustrated with pictures, mostly in color, that help explain the manifold properties described in the text. Each chapter ends with a set of exercises and explorations that both challenge the reader to prove assertions made in the text, and suggest further topics to explore that bring additional insight. There is an extensive index and bibliography.

Three-manifolds (Topology) --- Geometry, Hyperbolic. --- Complex manifolds. --- Hyperbolic spaces.

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Differential geometry. Global analysis --- Differentiable dynamical systems --- Hyperbolic spaces --- Invariant manifolds