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

Problem 1.3.
What can be said about the structure of subgroups of $F$?
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 quasiordered 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 $F$?
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 CantorBendixon 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 CantorBendixon rank of the space of Subgroups of $F$? 
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 piecewiseprojective homeomorphisms of $S^1$ that fix $\infty$.
Problem 1.8.
What is the subgroup structure of $H$?
Is there an infinite antichain 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?
Cite this as: AimPL: Amenability of discrete groups, available at http://aimpl.org/amenablediscrete.