Hyperbolically Embedded Subgroups of Acylindrically Hyperbolic Groups - Lecture 1

Explore the fascinating world of acylindrically hyperbolic groups through a comprehensive lecture focusing on hyperbolically embedded subgroups. Begin with fundamental definitions and examples before delving into the geometric and algebraic properties of these unique subgroups. Investigate connections to random walks, action extensions, quasicocycle extensions, and relative hyperbolicity. Discover how hyperbolically embedded subgroups, particularly those that are virtually cyclic, serve as powerful tools in proving significant results about acylindrically hyperbolic groups. Gain insights into these important results and understand the crucial role of hyperbolically embedded subgroups in their proofs. No prior knowledge of these topics is required, making this lecture an excellent starting point for those interested in advanced group theory and geometric group theory.