The geometry of biinvariant word metrics

Let G be a normally finitely generated group and let |g| be the
conjugation invariant word norm of an element g of G. We are interested in
the growth rate of the function n --> |g^n|. A priori this can be anything
between bounded and linear. I will show that for many classes of groups
there is a dichotomy: |g^n| is either bounded or linear. The groups for
which this is true include: braid groups, Coxeter groups, right-angled
Artin groups, mapping class groups of closed surfaces, lattices in simply
connected solvable Lie groups, Baumslag-Solitar groups, hyperbolic groups,
SL(n,Z) and some other lattices.

I know no finitely presented group which does not satisfy the above
dichotomy (there is a finitely generated example). I don't know
if the dichotomy holds for lattices in semisimple Lie groups.

This is a recent joint work with M.Brandenbursky, S.Gal and
M.Marcinkowski: http://arxiv.org/abs/1310.2921