Event № 460
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.
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 characterisation of (un)bounded elements. It follows for example, that if g is a simple separating curve on a surface, then |g^n| growths linearly. However, if g is a simple non-separating curve, then |g^n| <= 2 for every n. This answers a question of D. Calegari.
The main idea of the proof is to construct appropriate quasimorphisms. M. Abert asked if there are Aut-invariant nontrivial homogeneous quasimorphisms on free groups. As a by-product of our technique we answer this question in the positive for rank 2. This is a joint work with M. Brandenbursky.