Properties of Følner sequences for countably infinite, finitely generated, amenable, periodic/torsion groups

Question

I've managed to prove certain things about a class of groups, and the only remaining class of groups are those specified in the title. I'm mainly studying symbolic dynamics and not group theory, so I'm not very familiar with properties that these groups should have.

Specifically, my goal is to find some sort of construction of a shift on such groups, however to have any hope of doing so, I need to have an idea about what properties Følner sequences on such groups can be made to have.

For instance, if $G$ is such a group and $\{F_n\}_{n=1}^\infty$ is a right Følner sequence, then for any $g \in G$, is it the case that $\{\langle g\rangle F_n\}_{n=1}^\infty$ is also a right Følner sequence? Are such groups guaranteed to have sub-exponential growth, meaning the balls according to the word metric (associated with a generating set for $G$) form a Følner sequence?

At this point, I'm not entirely sure what the construction would look like, so I'm not even sure what properties I would want the Følner sequence to satisfy. I don't have any reasonable properties I know of at the moment that would allow me to begin thinking of such a construction, so any specific properties that you could think of would be helpful, since I might be able to find a way to use it in a construction. (Tractable) examples of such groups would also be helpful, since they may lead to a counterexample to what I'm attempting to prove.

Thanks in advance!

Answer 1

Converting my comment, i.e. construction of countably infinite, finitely-generated, amenable torsion group with exponential growth.

Pick any group $G$ which is countably infinite, finitely-generated, amenable and torsion, e.g. the Grigorchuk group. Then the (small) wreath product still has all the properties $\mathbb{Z}_2 \wr G$, since amenability and torsion are closed under group extensions.

The new group is of exponential growth, here's a direct embedding of a binary tree in a Cayley graph of $G$: Let $S$ be any generating set for $G$ and let $f$ be the generator that flips the bit at the origin. Pick using Kőnig's lemma a one-sided injective path $p \in S^\omega$, i.e. such that all prefixes $p_0 p_1 \cdots p_{n-1} \in S^n$ of $p$ evaluate to distinct elements of $G$. Now obviously the elements $f^{w_0} p_0 f^{w_1} p_1 \cdots f^{w_{n-1}} p_{n-1}$ are distinct for distinct binary words $w$, so we have embedded a binary tree over the generators $S \cup fS$.