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

TL;DR

This paper introduces hyperbolically embedded subgroups and rotating families, providing a unified framework to analyze various groups acting on hyperbolic spaces, leading to new results in mapping class groups and relatively hyperbolic groups.

Contribution

It develops the concepts of hyperbolically embedded subgroups and rotating families, generalizing existing structures and applying them to a broad class of groups, including new insights into mapping class groups.

Findings

Solved two open problems in mapping class groups

Obtained new results for relatively hyperbolic groups

Provided a framework applicable to groups acting on CAT(0) spaces

Abstract

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 later 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.