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

TL;DR

This paper explores the properties of equationally compact subgroups in countable groups, demonstrating the existence of subgroups with arbitrarily high Cantor-Bendixson ranks and constructing many weakly incomparable ergodic actions.

Contribution

It constructs examples of equationally compact subgroups with arbitrary Cantor-Bendixson ranks and many weakly incomparable Bernoulli shift actions, answering open questions.

Findings

Existence of equationally compact subgroups with any countable Cantor-Bendixson rank.

Construction of continuum many pairwise weakly incomparable ergodic actions.

Resolution of open questions on equational compactness by Prest and Rajani.

Abstract

Equationally compact subgroups of countable groups were introduced by Banaschewski. For all known cases the orbit closure of such a subgroup is a countable subset in the space of subgroups and has finite Cantor-Bendixson rank. We show that there exists a finitely generated group $\Gamma$ such that for 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 continuum many equationally compact subgroups of $\Gamma$ such that the associated ergodic Bernoulli shift actions are pairwise weakly incomparable. We also answer two questions on equational compactness posed by Prest and Rajani.