
## 2. Topological Full group, IET and PRG

1. #### Problem 2.1.

Is non-amenability of a topological full group preserved under taking restrictions?

To be more precise: let $G$ act on a Cantor set $C$ minimally. Assume $G$ is equal to its full group and is non-amenable.

Take a clopen subset $U$ of the Cantor set $C$, let $G_U$ be the restriction of $G$ to $U$. Is $G_U$ also non-amenable?
• #### Problem 2.2.

Is the group of $IET$ (interval exchange transformations) amenable?
• #### Problem 2.3.

Is the group of polygon rearrangements (PRG) amenable? Specifically, is the full group of the Penrose tiling amenable?
• #### Problem 2.4.

Does the topological full group of the (Morse, Chacon, Rudin-Shapiro) substitution subshift contain a subgroup of intermediate growth?
• #### Problem 2.5.

Find an example of a $\mathbf{Z}^2$-action on a Cantor set such that the commutator subgroup of the topological full group is f.p.
• #### Problem 2.6.

Are full groups of labeled graphs of "low" complexity (and "low" growth) amenable? (E.g. "low" = polynomial.)

What about different interpretations of complexity?
• #### Problem 2.7.

Take a hyperfinite (ergodic) equivalence relation on $[0,1]$. Let $G$ be the full group of all measure-preserving transformations that respect the equivalence relation (i.e., whose graph is contained in the equivalence relation).

Q: Does $G$ contain any discrete non-amenable subgroups?

Here, discrete means w.r.t. the distance $d(a,b) \mathrel{\mathop:}= \mu \{x\ :\ ax \neq bx \}$

Juschenko: If the answer is no, then IET is amenable.

Cite this as: AimPL: Amenability of discrete groups, available at http://aimpl.org/amenablediscrete.