Interval Exchange Transformations Groups: Free Actions and Dynamics of Virtually Abelian Groups

In this paper, we study groups acting freely by IETs. We first note that a finitely generated group admits a free IET action if and only if it is virtually abelian. Then, we classify the free actions of non virtually cyclic groups showing that they are "conjugate" to actions in some specific subgroups G n , namely G n ≃ (G 2) n ⋊ S n where G 2 is the group of circular rotations seen as exchanges of 2 intervals and S n is the group of permutations of {1, ..., n} acting by permuting the copies of G 2. We also study non free actions of virtually abelian groups and we obtain the same conclusion for any such group that contains a conjugate to a product of restricted rotations with disjoint supports and without periodic points. As a consequence, we get that the group generated by f ∈ G n periodic point free and g / ∈ G n is not virtually nilpotent. Moreover, we exhibit examples of finitely generated non virtually nilpotent subgroups of IETs, some of them are metabelian and others are not virtually solvable.