Solvable groups of interval exchange transformations

Abstract

We prove that any finitely generated torsion free solvable subgroup of the group $\mathrm{IET}$ of all Interval Exchange Transformations is virtually abelian. In contrast, the lamplighter groups $A\wr {\mathbb{Z}}^k$ embed in $\mathrm{IET}$ for every finite abelian group $A$, and we construct uncountably many non pairwise isomorphic 3-step solvable subgroups of $\mathrm{IET}$ 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 $\mathrm{IET}$.

17 pages, 2 figures, to appear

References in corpus (1)

• Extensive amenability and an application to interval exchanges

Cited by in corpus (1)

• Commensurating actions for groups of piecewise continuous transformations