Groups of interval exchange transformations
This is a joint work with Vincent Guirardel and Koji Fujiwara.
An interval exchange transformation is a bijective transformation
of an interval that consists in cutting it into finitely many
subintervals, and rearranging them by translations. These
transformations have been much studied as individual dynamical systems.
However, not much is known about the way they interact with each other.
The set of all interval exchange transformations is a group for the
composition. To investigate how two transformations interact with each
other, it is natural to ask what kind of group they generate. What can
appear as a subgroup of the full group of all interval exchange
transformations ? Is it possible to generate a free group ? Or may be
some other interesting group ? We investigate several cases. First, we
show that the only connected Lie groups that can be embedded in this
group are the abelian ones. Another result is that any subgroup of
interval exchange transformations with Kazhdan's property (T) is
finite. We investigate the possibilities for a free subgroup. We show
that in a natural model of genericity, and under a technical assumption
of irreducibility, a generic pair of interval exchange transformations
does not generate a free group.