Is the group of interval exchange transformations amenable?

Cut a circle into finitely many arcs and rearrange them by rotations: the result is an interval exchange transformation. These maps form a group, IET, that appears throughout dynamics, for instance in the study of polygonal billiards. Is this group amenable, so that it admits invariant averaging? Amenability is known for its subgroups of small "rational rank", but not in general.

Problem

Let $\mathbb{T} = \mathbb{R}/\mathbb{Z}$. An interval exchange transformation is a bijection $g : \mathbb{T} \to \mathbb{T}$ for which $\mathbb{T}$ splits into finitely many half-open intervals $[a, b)$ on each of which $g$ is a translation. Under composition these form the group IET, considered as a discrete group.

A discrete group $G$ is amenable if there is a finitely additive probability measure on all subsets of $G$ that is invariant under left translation; for IET this is equivalent to every finitely generated subgroup being amenable.

Question. Is IET amenable?

Background

The question is listed in the problem list of the AIM workshop on amenability of discrete groups. A closely related older question of Katok asks whether IET contains a free group on two generators; a free subgroup would make IET non-amenable. Interest grew after Juschenko and Monod (2013) gave the first examples of finitely generated infinite simple amenable groups, through an existence result for invariant states for piecewise translations of the integers.

What is known

• Juschenko, Matte Bon, Monod and de la Salle (2018) proved that every subgroup of IET whose translation amounts generate a subgroup of $\mathbb{T}$ of rational rank at most 2 is amenable, using extensive amenability; they also showed that some subgroups admit no finitely supported random walk with trivial Poisson boundary.
• Dahmani, Fujiwara and Guirardel (2013) showed that for a generic pair of interval exchanges the group they generate is not free; whether IET contains any free subgroup (Katok's question) is open.
• Juschenko, Matte Bon, Monod and de la Salle showed that IET would be amenable if, for every $d$, the action of the wobbling group of $\mathbb{Z}^d$ (permutations of $\mathbb{Z}^d$ moving points a bounded distance) on $\mathbb{Z}^d$ were extensively amenable. This was proved for $d = 1$ by Juschenko and Monod and for $d = 2$ by Juschenko, Nekrashevych and de la Salle; it is open for $d \ge 3$. Garban (2018) proved that a stronger property, diffuse extensive amenability, holds for such actions on a bounded-degree graph exactly when the graph is recurrent, so it fails for $\mathbb{Z}^d$ with $d \ge 3$; Garban conjectures that the wobbling action on $\mathbb{Z}^d$ is not extensively amenable when $d \ge 3$.

What would count as progress

• A proof that subgroups of rational rank 3 are amenable, or a non-amenable finitely generated subgroup of IET.
• An answer to Katok's question: two explicit interval exchanges generating a free group (whose relations, or absence of them, can be tested by computer on long words), or a proof that IET has no free subgroups.
• Extensive amenability of the wobbling group action on $\mathbb{Z}^d$ for $d \ge 3$, which would imply amenability of IET.

References

1. K. Juschenko, N. Matte Bon, N. Monod and M. de la Salle, Extensive amenability and an application to interval exchanges, Ergodic Theory Dynam. Systems 38 (2018), 195–219; arXiv:1503.04977.
2. K. Juschenko and N. Monod, Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. 178 (2013), 775–787.
3. K. Juschenko, V. Nekrashevych and M. de la Salle, Extensions of amenable groups by recurrent groupoids, Invent. Math. 206 (2016), 837–867.
4. F. Dahmani, K. Fujiwara and V. Guirardel, Free groups of interval exchange transformations are rare, Groups Geom. Dyn. 7 (2013), 883–910.
5. C. Garban, Inverted orbits of exclusion processes, diffuse-extensive-amenability and (non-?)amenability of the interval exchanges, arXiv:1804.01981 (2018).

Source

Imported from AIM Problem Lists amenablediscrete 2.2, curated by American Institute of Mathematics. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.