# What are the main open problems in the theory of amenability of groups?

I have read the Paterson and Runde books about amenability of groups, but I do not know what are the most intriguing questions in this area today.

A survey or a list of questions would be welcome.

• Notoriously difficult question about amenability: mathoverflow.net/questions/26821/is-thompsons-group-f-amenable Aug 18 at 16:07
• You can see the "Open problems" chapter of Volker Runde's book, "Lectures on amenability". Aug 18 at 17:24
• I'd say a general problem is to find amenable but not elementary amenable groups with interesting properties. For example, find such a group with a finite classifying space: mathoverflow.net/questions/53803/…. Or even one that's of type $FP_3$. (The Grigorchuk group is a finitely generated example, and there's a variation on it that's finitely presented, but for finiteness properties other than those, no one knows (as far as I know).) Aug 18 at 17:39
• @MeisamSoleimaniMalekan That book is mostly about amenability of Banach algebras and so that problem list has very little to say about amenability problems for groups Aug 19 at 20:38
• Just to reply to the wording of the original question: there is no Runde book about amenability of groups, but rather a book (and a significantly updated version) about amenability of Banach algebras. The first book did include some outline of Banach-Tarski and hence of non-amenability of F_2 Aug 19 at 20:42

$$\DeclareMathOperator\IET{IET}$$ A folklore question is the amenability of the group $$\IET$$ of interval exchanges transformations (= right continuous permutations of the interval $$[0,1\mathclose[$$ that locally coincide with translations outside a finite subset).

(Unlike Thompson's group $$F$$, it is also unknown whether this group contains a non-abelian free subgroup; this question is attributed to A. Katok.)

Progress on this question: for obvious reasons, amenability of $$\IET$$ is equivalent to amenability of all its finitely generated subgroup. For a subgroup $$G$$ of $$\IET$$, define its rank $$r(G)$$ as the $$\mathbf{Q}$$-rank of the subgroup $$\Lambda_G$$ of $$\mathbf{R}/\mathbf{Z}$$ generated by translation lengths of elements of $$G$$ modulo $$\mathbf{Z}$$. If $$G$$ is finitely generated, then $$r(G)<\infty$$.

For instance $$r(G)=0$$ means that $$G$$ has only rational translation lengths, and is then locally finite, hence (obviously) amenable.

If $$r(G)\le 1$$ then $$G$$ is amenable: this is due to Juschenko and Monod (technically, they need $$\Lambda_G$$ to be cyclic and not just virtually cyclic, but basically this is what their method, not specific to $$\IET$$, provides). Using random walks, Juschenko, Matte Bon, Monod and de la Salle proved it for $$r(G)\le 2$$. The recurrence of the planar random walk is used in the proof, so $$r(G)=3$$ is unknown at this stage, as well as the general case.

I don't know if there's any conjecture about this question. I tend to bet on a positive answer ($$\IET$$ is amenable) but have no evidence enough to conjecture it.

• I believe the question of amenability of the IET group is due to Koji Fujiwara. Aug 18 at 19:47
• @UriBader do you have a written reference? I think I asked the question at least since around 2010 (when I heard about the free group question, possibly in a talk about the Dahmani-Fujiwara-Guirardel ongoing work on free groups in IET). In this setting the question comes so naturally that it has probably been asked by many.
– YCor
Aug 18 at 20:06
• ... and in 2011 I remember making an unsuccessful attempt to adapt the Grigorchuk-Medynets paper to prove amenability of IET (later the proof there was deemed not correct).
– YCor
Aug 18 at 20:12
• I heard a talk by Koji at the Technion (I don't remember the year) where he mentioned Katok's conjecture and said that as far as he knows the group could be amenable. I have no further information. Aug 19 at 5:55

Does there exist an infinite finitely presented simple amenable group?

Weaker variant: does there exist an infinite finitely presented aperiodic amenable group? Here aperiodic means: no nontrivial finite quotient.

The same problems for finitely generated groups were solved about 10 years ago. The final step was due to Juschenko and Monod, who proved the amenability of topological-full groups of Cantor minimal $$\mathbf{Z}$$-subshifts (2012, published 2013). Previously Matui proved that their derived subgroup is simple and finitely generated (2006). That these groups, previously introduced by Putnam / Giordano-Putnam-Skau in the 90s, are amenable, was conjectured by Grigorchuk and Medynets in 2011.

However these groups (as well as IET) are LEF and hence their finitely presented subgroups are residually finite, so there's not even a plausible candidate at this point, as far as I know.

(I'm not sure of a reference for this question: possibly it can be found in the Baumslag or Kourovka list?)

The Part C of Appendices of the book "Amenability of discrete groups by examples" by Kate Juschenko, contains open problems in amenability.