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1. Thompson group F and groups of homeomorphisms of the interval and the circle


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      Problem 1.1.

      Is the Thompson group F amenable?


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          Problem 1.2.

          Is F sofic?


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              Problem 1.3.

              What can be said about the structure of subgroups of F?
                1. Remark. Bleak, Brin and Moore proved that there is a chain of length \epsilon_0 of elementary amenable subgroups of F (ordered with respect to embeddability) where \epsilon_0 is the smallest ordinal such that \epsilon_0=\omega^{\epsilon_0}
                    • Remark. [Bleak, Brin and Moore] There is a chain of length \epsilon_0 of elementary amenable subgroups of F (ordered with respect to embeddability) where \epsilon_0 is the smallest ordinal such that \epsilon_0=\omega^{\epsilon_0}

                        •     We say that a collection of groups \mathcal C is quasi-ordered if for any sequence \{G_i\}_{i\in\mathbb{N}} of groups in \mathcal C there is i< j such that G_i embeds into G_j.

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                          Problem 1.4.

                          Is it true that a subgroup of F is elementary amenable if and only if it does not contain a copy of F?
                            1. Remark. It is conjectured by Brin and Sapir that the answer is positive.

                                •     The derivative of a subset S of a topological space X, denoted S', is the set of limit points of S. By transfinite induction, one defines the \alpha-derivative of S for every ordinal \alpha. The Cantor-Bendixon rank of S is the smallest ordinal \beta such that the \beta-derivative of S and the (\beta+1)-derivative of S coincide.

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                                  Problem 1.5.

                                  What is the Cantor-Bendixon rank of the space of Subgroups of F?
                                      The Cantor-Bendixon Rank of F is at least \omega. Proved during the workshop by Elder, Grigorchuk, Kassabov, Moore and Wesolek.


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                                      Problem 1.6.

                                      Is there an invariant continuous probability measure for the action of F acting on its perfect kernel (by conjugation)?


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                                          Problem 1.7.

                                          What are the maximal subgroups of F, and specifically, are there elementary amenable maximal ones?

                                            •     Let H be the group of all piecewise-projective homeomorphisms of S^1 that fix \infty.

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                                              Problem 1.8.

                                              What is the subgroup structure of H?

                                              Is there an infinite anti-chain of subgroups of H?

                                              Are there only finitely many obstructions to amenability? (i.e., is there a finite collection of subgroups of H such that every non amenable subgroup of H contains a copy of one of them?) This question is also interesting for subgroups of Thompson’s group F or more generally, for subgroups of \mathrm{PL}_o(I) (the group of piecewise linear orientation preserving homeomorphisms of the interval [0,1] with finitely many breakpoints).

                                                •     We say that a group G is invariably generated by \{x_1,\dots,x_n\} if for any g_1, \ldots, g_n \in F, \{g_1^{-1}x_1g_1, \ldots, g_n^{-1}x_ng_n\} generates G.

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                                                  Problem 1.9.

                                                  Is F is invariably generated by a finite set?
                                                      Solved. F is invariably generated by a finite set (Golan and Juschenko).

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