Some elementary amenable subgroups of interval exchange transformations

In this paper, we study a family of finitely generated elementary amenable iet-groups. These groups are generated by finitely many rationals iets and rotations. For them, we state criteria for not virtual nilpotency or solvability, and we give conditions to ensure that they are not virtually solvable. We precise their abelianizations, we determine when they are isomorphic to certain lamplighter groups and we provide non isomorphic cases among them. As consequences, in the class of infinite finitely generated subgroups of iets up to isomorphism, we exhibit infinitely many non virtually solvable and non linear groups, and infinitely many solvable groups of arbitrary derived length.

💡 Research Summary

This paper investigates a broad family of finitely generated subgroups of the interval exchange transformation group (IET). Each group is built from a finite set of irrational rotations together with a finite collection of rational IETs that act as permutations on a fixed rational partition of the unit interval. Formally, for a set of Q‑independent irrational numbers (\alpha_{1},\dots,\alpha_{s}) let (A=\langle\alpha_{i}\rangle\cong\mathbb Z^{s}) and let (Q=\langle g_{1},\dots,g_{m}\rangle) be a finitely generated subgroup of the finite permutation group (G_{1,q}) (the image of the symmetric group (S_{q}) inside IET). The main object of study is

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