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Maximal cofinitary groups revisited. (English) Zbl 1372.03096

Summary: Let \(\kappa\) be an arbitrary regular infinite cardinal and let \(\mathcal C\) denote the set of \(\kappa\)-maximal cofinitary groups. We show that if \(\mathsf {GCH}\) holds and \(\mathcal C\) is a closed set of cardinals such that
1.
\(\kappa + \in C\), \(\forall\nu\in C(\nu \geq \kappa^ + )\),
2.
if \(|C| \geq \kappa ^+\) then \([\kappa^ + , |C|] \subseteq C\),
3.
\( \forall \nu \in C(\mathrm{cof}(\nu) \leq \kappa\to \nu ^+ \in C)\),
then there is a generic extension in which cofinalities have not been changed and such that \(C = \{|\mathcal G| :\mathcal G \in \mathcal C\}\). The theorem generalizes a result of J. Brendle et al. [J. Algebra 232, No. 1, 209–225 (2000; Zbl 0966.20001)] regarding the possible sizes of maximal cofinitary groups.
Our techniques easily modify to provide analogous results for the spectra of maximal \(\kappa\)-almost disjoint families in \([\kappa]^ \kappa \), maximal families of \(\kappa\)-almost disjoint permutations on \(\kappa\) and maximal families of \(\kappa\)-almost disjoint functions in \({}^\kappa\kappa\). In addition we construct a \(\kappa\)-Cohen indestructible \(\kappa\)-maximal cofinitary group and so establish the consistency of \(\mathfrak a _g (\kappa) < \mathfrak d(\kappa)\), which for \(\kappa =\omega\) is due to Y. Zhang [Arch. Math. Logic 39, No. 1, 41–52 (2000; Zbl 0954.20001)].

MSC:

03E35 Consistency and independence results
20B07 General theory for infinite permutation groups
03F05 Cut-elimination and normal-form theorems
03E17 Cardinal characteristics of the continuum
20A15 Applications of logic to group theory
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References:

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