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4. Amenability+

Questions pertaining to extensions, relexations, and generalizations of the notions of amenability and related properties (such as the Liouville property). Amenability in conjuction with other properties (is there an amenable ______ group?).

    1.     Definition: A group $G$ is supramenable if for every triplet $(G,X,E)$ (where $E$ is a nonempty subset of $X$ and $G \def\actson{\curvearrowright} X$), there is an invariant measure normalized on $E$.

      Fact: Subexponential growth implies supramenability.

      Add remark

      Problem 4.05.

      1. Q: (Rosenblatt 74’) Is there a supramenable group of exponential growth?
      2. Is the direct product of $2$ supramenable groups also supramenable?

        •     Definition: A hereditary just-infinite group $G$ is a residually finite group such that every finite index subgroup of it is just-infinite.

          Add remark

          Problem 4.1.

          Is there an amenable hereditary just-infinite group that is not elementary amenable? Is there a finitely presented one?


            • Add remark

              Problem 4.15.

              Is there a f.p. amenable infinite simple group? A construction of Juschenko and Monod (2012) gives a finitely generated infinite simple amenable group. Conjecture: The derived subgroup of the full group of the Penrose tiling is a candidate.


                • Add remark

                  Problem 4.2.

                  Is there an infinite finitely generated amenable simple non-Liouville group (with respect to symmetric, finitely supported, generating measures)?

                    •     Definition: SQ-closure = closure under subgroups and quotients.

                      Add remark

                      Problem 4.25.

                      1. What are "good" SQ-closed families of amenable groups?
                      2. What is the SQ-closure of the set of bounded automata groups?


                        • Add remark

                          Problem 4.3.

                          Are group actions with polynomial-growth Schreier graphs extensively amenable?

                          Extensive amenability is known for recurrent actions.

                            •     Definition: let AE be the smallest class of locally compact groups s..t. the following holds:
                              1. AE contains all compact groups and AE contains all discrete amenable groups;
                              2. closed under the (top) elementray ops.

                              Add remark

                              Problem 4.35.

                              1. is every ever locally compact amenable group in AE?
                              2. Find f.g. amenable groups with infinite commensurated subgroups with trivial normal core.
                              3. Is every commensurated subroup of the Basilica group commensurate with a normal subgroup?


                                • Add remark

                                  Problem 4.4.

                                  Is there a non-amenable group that has a Liouville action with amenable stabilizers?


                                    • Add remark

                                      Problem 4.45.

                                      Is there a transitive action $G \def\actson{\curvearrowright} X$ that is $\mu$-Liouville, but the action on some orbit of $G\def\actson{\curvearrowright} \subset X \times X$ is not $\mu^2$-Liouville?


                                        • Add remark

                                          Problem 4.5.

                                          Is there a transitive action $G \def\actson{\curvearrowright} X$ that is $\mu$-Liouville, but the action on some orbit of $G\def\actson{\curvearrowright} \subset X \times X$ is not $\mu^2$-Liouville?


                                            • Add remark

                                              Problem 4.55.

                                              Are contracting groups amenable?


                                                • Add remark

                                                  Problem 4.6.

                                                  Is the Liouville property stable under the choice of of symmetric, finitely supported, generating measure?


                                                    • Add remark

                                                      Problem 4.65.

                                                      Is there a 2-generated infinite simple amenable group?


                                                        • Add remark

                                                          Problem 4.7.

                                                          Is there a 2-generated infinite simple amenable group?


                                                            • Add remark

                                                              Problem 4.75.

                                                              Is there an amenable group whose spectrum is a Cantor set?

                                                              Remark (Dudko): For Grigorchuk’s group $G$, the spectrum is a union of two intervals: $[-1,0]\cup[1/2,1]$; the question is whether there can be something a tad crazier.

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