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3. Grigorchuk’s group, branch groups and groups of intermediate growth


    1. Add remark

      Problem 3.1.

      Is there a f.p. branch group? Is there a f.p. amenable branch group?

        •     Def: A subgroup H of a group G is commensurated if \forall g \in G, g^{-1}Hg \cap H has finite index in H.

          Thm(Wesolek): A finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index.

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          Problem 3.2.

          Find a combinatorial proof of the following: every commensurated subgroup of a finitely generated branch group is either finite or of finite index.
              The problem is interesting even in the special case of Grigorchuk’s group G


            • Add remark

              Problem 3.3.

              Is the universal Grigorchuk group amenable?
                1. Remark. If this group is amenable, then the Folner function of this group is universal bound on Folner function of G_\omega


                    • Add remark

                      Problem 3.4.

                      Find bounds on Folner functions for the Grigorchuk group G.


                        • Add remark

                          Problem 3.5.

                          Is the sequence of balls B_n(e) in Grigorchuk’s group G a Folner sequence?


                            • Add remark

                              Problem 3.6.

                              Does the limit \lim_{n\rightarrow\infty} \frac{\log \log (B_G(n))}{\log n}
                              exist for the Grigorchuk group?


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                                  Problem 3.7.

                                  Is \rho_c<1 for all groups of intermediate growth?


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                                      Problem 3.8.

                                      Is there a finitely presented amenable group that surjects onto a group of intermediate growth?


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                                          Problem 3.9.

                                          Are polynomial activity automata groups amenable?

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