Abstract:  In this talk I will address three following very important and apparently unrelated questions:

1. Given 2 groups H and G, does there exists a monomorphism from H to G? In other words can one describe subgroups of a given group G?

2. Given 2 groups H and G, does there exists an epimorphism from G to H? In other words can one describe quotients of a given group G?

3. Given a diffeomorphism of a smooth manifold, can one factor it into "simple" pieces?

The goal of my talk is to to convince you that for many groups of geometric origin (e.g. groups of diffeomorphisms of smooth manifold which do or do not preserve some additional structure) these questions are related. More precisely, I will discuss a strategy which can be applied to all these questions simultaneously.