Reversible maps in groups of interval exchange transformations

Guelman, Nancy; Liousse, Isabelle

doi:10.2140/agt.2026.26.2497

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author index

To appear

Other MSP journals

Reversible maps in groups of interval exchange transformations

Nancy Guelman and Isabelle Liousse

Algebraic & Geometric Topology 26 (2026) 2497–2518

DOI: 10.2140/agt.2026.26.2497

Abstract

An element $f$ of a group $G$ is reversible if it is conjugated in $G$ to its own inverse. It is plain that two elements $f$ and $h$ satisfying the relation $h f h^{- 1} = f^{- 1}$ generate a, possibly nonfaithful, $BS (1, - 1)$ -action, where $BS (1, - 1)$ is the Baumslag–Solitar group $⟨ a, b ∣ b a b^{- 1} = a^{- 1} ⟩$ also referred to as the Klein bottle group.

Our main result is that any reversible interval exchange transformation (IET) is reversible by a finite order IET. Its proof uses a dynamical decomposition for $BS (1, - 1)$ -actions by IET which involves some specific IET-subgroups for which we give a precise description of the reversible elements. We also indicate consequences on the IET-realizations of a group containing reversible maps.

Keywords

interval exchange transformation, reversible map, involution

Mathematical Subject Classification

Primary: 37E05, 57S30

Secondary: 37B05, 37C85

References

Publication

Received: 19 June 2024

Accepted: 29 June 2025

Published: 1 September 2026

Authors

Nancy Guelman
Instituto de Matemática y Estadistica Rafael Laguardia Universidad de la República Montevideo Uruguay

Isabelle Liousse
Université de Lille 1 CNRS, UMR 8524 - Laboratoire Paul Painlevé Lille France

Open Access made possible by participating institutions via Subscribe to Open.

Volume 26, issue 7 (2026)