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

1. #### Problem 1.1.

Is the Thompson group $F$ amenable?
• #### Problem 1.2.

Is $F$ sofic?
• #### 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 $G_j$.

#### 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.

#### 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.
• #### Problem 1.6.

Is there an invariant continuous probability measure for the action of $F$ acting on its perfect kernel (by conjugation)?
• #### 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 $\infty$.

#### 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 $G$.

#### 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.