Interval exchanges that do not occur in free groups

Abstract

A disjoint rotation map is an interval exchange transformation (IET) on the unit interval that acts by rotation on a finite number of invariant subintervals. It is currently unknown whether the group $\mathcal{E}$ of all IETs possesses any non-abelian free subgroups. It is shown that it is not possible for a disjoint rotation map to occur in a subgroup of $\mathcal{E}$ that is isomorphic to a non-abelian free group.