Denis Osin, City College of New York

Abstract I will concentrate on the following two well known questions.

Problem 1. Is every hyperbolic group residually finite?

Problem 2. Are peripheral subgroups of a finitely presented relatively hyperbolic group finitely presented?

The first problem can be stated in many equivalent forms in terms of hyperbolic, relatively hyperbolic, or even periodic groups. The second one can be approached via asymptotic cones and has positive solution for groups with quadratic Dehn functions. These questions have a striking connection that leads to interesting conjectures about group embeddings.