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2. Topological Full group, IET and PRG


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      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?


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          Problem 2.2.

          Is the group of $IET$ (interval exchange transformations) amenable?


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              Problem 2.3.

              Is the group of polygon rearrangements (PRG) amenable? Specifically, is the full group of the Penrose tiling amenable?


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                  Problem 2.4.

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


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                      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.


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                          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?


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                              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.