On conjugation invariant geometry of groups

I will discuss basic properties of conjugation invariant norms on groups. I will explain why some groups are bounded (a group G is bounded if every conjugation invariant norm on G has finite diameter). The examples will include diffeomorphism groups of manifolds and some lattices in semisimple Lie groups. I will discuss the stronger notion of uniform boundedness. I will also provide examples of unbounded groups, some open problems and an application to group actions on manifolds.