Aut-invariant metrics and Aut-invariant quasimorphisms on free groups and surface groups.

**Lecture 2. **

There are two interesting norms on free groups and surface groups which are invariant under the group of all automorphisms:

A) For free groups we have the primitive norm, i.e., |g|_p = the minimal number of primitive elements one has to multiply to get g.

B) For fundamental group of genus g surface we have the simple curves norm, i.e., |g|_s = the minimal number of simple closed curves one need to concatenate to get g.

In our recent paper with M. Brandenbursky we prove the following dichotomy: either |g^n| is bounded or growths linearly with n. For free groups and surface groups we give an explicit characterization of (un)bounded elements.

In two talks I will explain the idea of the proof and draw a number of consequences. The proof uses the theory of mapping class groups (i.e. Nielsen-Thurston normal form, Birman embedding) and quasimorphisms.