2. Topological Full group, IET and PRG

Problem 2.1.
Is nonamenability 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 nonamenable.
Take a clopen subset $U$ of the Cantor set $C$, let $G_U$ be the restriction of $G$ to $U$. Is $G_U$ also nonamenable? 
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, RudinShapiro) 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 measurepreserving transformations that respect the equivalence relation (i.e., whose graph is contained in the equivalence relation).
Q: Does $G$ contain any discrete nonamenable 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.