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 \(G_n \simeq (\mathcal {G}_2)^{n}\rtimes \mathcal S_{n}\) where \(\mathcal {G}_2\) is the group of circular rotations seen as exchanges of 2 intervals and \(\mathcal S_{n}\) is the group of permutations of \(\{1,...,n\}\) acting by permuting the copies of \(\mathcal {G}_2\). 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 \(f\in G_n\) periodic point free and \(g\notin G_{n}\) 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.