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Some elementary amenable subgroups of interval exchange transformations

Nancy Guelman, Isabelle Liousse

TL;DR

The paper develops a family of finitely generated elementary amenable subgroups of IET, denoted $H_{A,Q}$, built from rotations by a free abelian group $A\cong \mathbb{Z}^s$ and finitely generated rational IETs $Q$. It introduces a morphism $\ell: H_{A,Q}\to \mathbb{R}/\mathbb{Z}$ capturing irrational translation data and analyzes the kernel via local permutations $\omega_x$, connecting the finite permutation actions to the global group structure. The authors establish sharp criteria for (non)virtual solvability, provide a detailed description of the abelianization, and show when $H_{A,Q}$ is isomorphic to lamplighter groups or to more general solvable or non-solvable types, yielding infinitely many nonisomorphic infinite subgroups of IET with varied solvability and growth. They also demonstrate that many $H_{A,Q}$ give rise to non virtually solvable, non-linear groups, while others yield solvable groups of arbitrary derived length, thereby enriching the landscape of finitely generated subgroups of IET with new, explicit examples. The work contributes tools (like $\ell$ and local permutations) that facilitate analyzing IET subgroups and their relations to wreath products and lamplighters, with implications for amenability and linearity questions in this setting.

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.

Some elementary amenable subgroups of interval exchange transformations

TL;DR

The paper develops a family of finitely generated elementary amenable subgroups of IET, denoted , built from rotations by a free abelian group and finitely generated rational IETs . It introduces a morphism capturing irrational translation data and analyzes the kernel via local permutations , connecting the finite permutation actions to the global group structure. The authors establish sharp criteria for (non)virtual solvability, provide a detailed description of the abelianization, and show when is isomorphic to lamplighter groups or to more general solvable or non-solvable types, yielding infinitely many nonisomorphic infinite subgroups of IET with varied solvability and growth. They also demonstrate that many give rise to non virtually solvable, non-linear groups, while others yield solvable groups of arbitrary derived length, thereby enriching the landscape of finitely generated subgroups of IET with new, explicit examples. The work contributes tools (like and local permutations) that facilitate analyzing IET subgroups and their relations to wreath products and lamplighters, with implications for amenability and linearity questions in this setting.

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.
Paper Structure (39 sections, 26 theorems, 23 equations, 1 figure)

This paper contains 39 sections, 26 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on iets and subgroups of rational iets
  3. Basic properties of $H_{A,Q}$
  4. The morphism $\ell$ and its kernel
  5. Generating sets for $\ker\ell$ and $[H,H]$
  6. Proofs of Theorem \ref{['THGenF1']}
  7. Local permutations
  8. Definitions and basic properties
  9. Images of $\ker \ell$ and $[H,H]$ by the local permutations.
  10. Proof of Theorem \ref{['THF0']}
  11. A criterium of non virtual solvability
  12. Proof of Item (1) of Theorem \ref{['THF0']}
  13. Proof of Item (2) of Theorem \ref{['THF0']}.
  14. Proofs of Theorem \ref{['THRFin']} and Corollaries \ref{['CPAb']} and \ref{['Noniso']}
  15. Proofs of Items (1) and (2) of Theorem \ref{['THRFin']}
  16. ...and 24 more sections

Key Result

Theorem 1

Let $A$, $Q$ be as in Definition defH and $\sigma=\sigma_q$ be the $q$-cycle $(1,2,...,q) \in \mathcal{S}_q$. Then either:

Figures (1)

  • Figure :

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 43 more