Interval Exchange Transformations groups. Free actions and dynamics of virtually abelian groups

Abstract

H\"older's theorem states that any group acting freely by circle homeomorphisms is abelian, this is no longer true for interval exchange transformations: we first give examples of free actions of non abelian groups. Then after noting that finitely generated groups acting freely by IET are virtually abelian, we classify the free actions of groups containing a copy of $\mathbb Z^2$, 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 provide examples of non virtually nilpotent subgroups of IETs. In particular, we show 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.