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This is crossposted from MSE.

We say a subset $A$ of a group $G$ is a monotile for $G$ if $G$ is a disjoint union of right translates of $A$.

In his article Monotileable Amenable Groups, B. Weiss gives lots of examples of amenable groups which admit a left-Følner sequence of monotiles (e.g. profinite or solvable groups). Is there an example of a countable amenable group which is known not to have a left-Følner sequence of monotiles?

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    $\begingroup$ While waiting for experts to answer: I think the shapes of Følner sets for topological full groups of minimal subshifts aren't known in a very concrete sense, so maybe those? $\endgroup$
    – Ville Salo
    Commented Apr 21 at 18:42
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    $\begingroup$ I think that this question is still open. It is also not yet known whether all amenable groups are monotileable. $\endgroup$
    – Ashot Minasyan
    Commented Apr 30 at 9:32

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