Hyperbolically Embedded Subgroups and Rotating Families in Groups Acting on Hyperbolic Spaces

F. Dahmani · V. Guirardel · D. Osin

Jan. 2017 · American Mathematical Soc.

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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.

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F. Dahmani: Université Grenoble Alpes, Grenoble, France,
V. Guirardel: Université de Rennes, France,
D. Osin: Vanderbilt University, Nashville, TN