Solvable groups of interval exchange transformations

TL;DR

This paper investigates the structure of solvable subgroups within the group of all Interval Exchange Transformations, establishing conditions for when these subgroups are virtually abelian and exploring embeddings of lamplighter groups.

Contribution

It proves that finitely generated torsion free solvable subgroups of IET are virtually abelian and constructs numerous non-isomorphic 3-step solvable subgroups, expanding understanding of subgroup embeddings.

Findings

Finitely generated torsion free solvable subgroups of IET are virtually abelian.

Lamplighter groups $A\wr \mathbb{Z}^k$ embed in IET for finite abelian A.

Non-abelian finite groups $F$ do not embed in IET.

Abstract

We prove that any finitely generated torsion free solvable subgroup of the group ${\rm IET}$ of all Interval Exchange Transformations is virtually abelian. In contrast, the lamplighter groups $A\wr \mathbb{Z}^k$ embed in ${\rm IET}$ for every finite abelian group $A$, and we construct uncountably many non pairwise isomorphic 3-step solvable subgroups of ${\rm IET}$ as semi-direct products of a lamplighter group with an abelian group. We also prove that for every non-abelian finite group $F$, the group $F\wr \mathbb{Z}^k$ does not embed in ${\rm IET}$.