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Almost finiteness was introduced in the seminal work of Kerr as an dynamicalanalogue of Z-stability in the Toms-Winter conjecture. In this article, weprovide the first examples of minimal, topologically free actions of amenablegroups that have mean dimension zero but are not almost finite. More precisely,we prove that there exists an infinite family of amenable wreath products thatadmit topologically free, minimal profinite actions on the Cantor space whichfail to be almost finite. Furthermore, these actions have dynamical comparison.This intriguing new phenomenon shows that Kerr's dynamical analogue ofToms-Winter conjecture fails for minimal, topologically free actions ofamenable groups.  The notion of allosteric group holds a significant position in our study. Agroup is allosteric if it admits a minimal action on a compact space with aninvariant ergodic probability measure that is topologically free but notessentially free. We study allostery of wreath products and provide the firstexamples of allosteric amenable groups.

Amenable wreath products with non almost finite actions of mean dimension zero