2023/05/08 by Nancy Guelman, Guelman, Nancy, Isabelle Liousse +1
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Cellular Automata and Applications #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.2305.04642
Hölder'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 Z2, showing that they are ``conjugate" to actions in some specific subgroups Gn, namely Gn ≃ (\mathcal G2)n \rtimes\mathcal Sn where \mathcal G2 is the group of circular rotations seen as exchanges of 2 intervals and \mathcal Sn is the group of permutations of \1,...,n\ acting by permuting the copies of \mathcal G2. 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∈ Gn periodic point free and g∉ Gn 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.