Some elementary amenable subgroups of interval exchange transformations

TL;DR

This paper investigates finitely generated elementary amenable subgroups of interval exchange transformations, providing criteria for their algebraic properties, classifying their structures, and demonstrating the diversity of their solvability and linearity.

Contribution

It introduces new criteria for non-virtual nilpotency and solvability, classifies certain subgroup structures, and shows the existence of diverse non virtually solvable and solvable groups within this class.

Findings

Identifies criteria for non-virtual nilpotency and solvability.

Classifies when these groups are isomorphic to lamplighter groups.

Shows existence of infinitely many non virtually solvable and solvable groups.

Abstract

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.