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 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 that is isomorphic to a non-abelian free group.
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Christopher F. Novak, Interval exchanges that do not occur in free groups. Groups Geom. Dyn. 6 (2012), no. 4, pp. 755–763DOI 10.4171/GGD/173