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

1. #### 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.

#### 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 $G$
• #### 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 $G_\omega$
• #### Problem 3.4.

Find bounds on Folner functions for the Grigorchuk group $G$.
• #### 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 3.7.

Is $\rho_c<1$ for all groups of intermediate growth?
• #### Problem 3.8.

Is there a finitely presented amenable group that surjects onto a group of intermediate growth?
• #### Problem 3.9.

Are polynomial activity automata groups amenable?

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