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

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.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. 
Problem 3.3.
Is the universal Grigorchuk group amenable?
Remark. If this group is amenable, then the Folner function of this group is universal bound on Folner function of $G_\omega$


Problem 3.5.
Is the sequence of balls $B_n(e)$ in Grigorchuk’s group $G$ a Folner sequence? 
Problem 3.6.
Does the limit \[ \lim_{n\rightarrow\infty} \frac{\log \log (B_G(n))}{\log n} \] exist for the Grigorchuk group? 
Problem 3.8.
Is there a finitely presented amenable group that surjects onto a group of intermediate growth?
Cite this as: AimPL: Amenability of discrete groups, available at http://aimpl.org/amenablediscrete.