Groups of interval exchange transformations
Fran├žois DAHMANI
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.