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Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces

About this Title

F. Dahmani, Université de Grenoble Alpes, Institut Fourier, F-38000 Grenoble, France, V. Guirardel, Université de Rennes 1 263 avenue du Général Leclerc, CS 74205, F-35042 RENNES Cedex France and D. Osin, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 245, Number 1156
ISBNs: 978-1-4704-2194-6 (print); 978-1-4704-3601-8 (online)
Published electronically: July 14, 2016
MSC: Primary 20F65, 20F67; Secondary 20F06, 20E08, 57M27

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Table of Contents


  • 1. Introduction
  • 2. Main results
  • 3. Preliminaries
  • 4. Generalizing relative hyperbolicity
  • 5. Very rotating families
  • 6. Examples
  • 7. Dehn filling
  • 8. Applications
  • 9. Some open problems


We 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. We obtain a number of general results about rotating families and hyperbolically embedded subgroups; although our technique applies to a wide class of groups, it is capable of producing new results even for well-studied particular classes. For instance, we solve two open problems about mapping class groups, and obtain some results which are new even for relatively hyperbolic groups.

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