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Equationally compact subgroups of countable groups were introduced byBanaschewski. For all known cases the orbit closure of such a subgroup is acountable subset in the space of subgroups and has finite Cantor-Bendixsonrank. We show that there exists a finitely generated group $\Gamma$ such thatfor any countable ordinal $\alpha$ we have an equationally compact subgroup$H\subset \Gamma$ for which the Cantor-Bendixson rank of the orbit closure of$H$ equals to $\alpha+2$. Then we give an explicite construction of continuummany equationally compact subgroups of $\Gamma$ such that the associatedergodic Bernoulli shift actions are pairwise weakly incomparable. We alsoanswer two questions on equational compactness posed by Prest and Rajani.

Invariant subsets of the space of subgroups, equational compactness and the weak equivalence of actions