Relatively Hyperbolic Groups Intrinsic Geometry Algebraic Properties And Algorithmic Problems

In this the authors obtain an isoperimetric characterization of relatively hyperbolicity of a groups with respect to a collection of subgroups. This allows them to apply classical combinatorial methods related to van Kampen diagrams to obtain relative analogues of some well-known algebraic and geometric properties of ordinary hyperbolic groups. There is also an introduction and study of the notion of a relatively quasi-convex subgroup of a relatively hyperbolic group and solve somenatural algorithmic problems.

Ten high-quality survey articles provide an overview of important recent developments in the mathematics surrounding negative curvature.

he 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, , and the Cremona group. Other examples can be found among groups acting geometrically on 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.

The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.

The purpose of this volume and of the other volumes in the same series is to provide a collection of surveys that allows the reader to learn the important aspects of William Thurston’s heritage. Thurston’s ideas have altered the course of twentieth century mathematics, and they continue to have a significant influence on succeeding generations of mathematicians. The topics covered in the present volume include com-plex hyperbolic Kleinian groups, Möbius structures, hyperbolic ends, cone 3-manifolds, Thurston’s norm, surgeries in representation varieties, triangulations, spaces of polygo-nal decompositions and of singular flat structures on surfaces, combination theorems in the theories of Kleinian groups, hyperbolic groups and holomorphic dynamics, the dynamics and iteration of rational maps, automatic groups, and the combinatorics of right-angled Artin groups.

Contains expository articles and research papers in geometric group theory focusing on generalisations of Gromov hyperbolicity.

The articles in this volume are based on the talks given at two special sessions at the AMS Sectional meetings held in 2004. The articles cover various topological and asymptotic aspects of group theory, such as hyperbolic and relatively hyperbolic groups, asymptotic cones, Thompson's group, Nielsen fixed point theory, homology, groups acting on trees, groups generated by finite automata, iterated monodromy groups, random walks on finitely generated groups, heat kernels, and currents on free groups.

Beginning by introducing a geometric mechanism for diffusion in a priori unstable nearly integrable dynamical systems. This book is based on the observation that resonances, besides destroying the primary KAM tori, create secondary tori and tori of lower dimension. It argues that these objects created by resonances can be incorporated in transition chains taking the place of the destroyed primary KAM tori.The authors establish rigorously the existence of this mechanism in a simplemodel that has been studied before. The main technique is to develop a toolkit to study, in a unified way, tori of different topologies and their invariant manifolds, their intersections as well as shadowing properties of these bi-asymptotic orbits. This toolkit is based on extending and unifyingstandard techniques. A new tool used here is the scattering map of normally hyperbolic invariant manifolds.The model considered is a one-parameter family, which for $\varepsilon = 0$ is an integrable system. We give a small number of explicit conditions the jet of order $3$ of the family that, if verified imply diffusion. The conditions are just that some explicitely constructed functionals do not vanish identically or have non-degenerate critical points, etc.An attractive feature of themechanism is that the transition chains are shorter in the places where the heuristic intuition and numerical experimentation suggests that the diffusion is strongest.