The structure of κ-maximal cofinitary groups.

Area of Science:

• Set Theory
• Group Theory
• Combinatorial Set Theory

Background:

• The study revisits Kastermans' work on kappa-maximal cofinitary groups.
• It builds upon a recently developed higher analogue of Bell's theorem.
• Focuses on groups where kappa is a regular uncountable cardinal satisfying kappa = 2^kappa.

Purpose of the Study:

• To characterize the number of orbits of kappa-maximal cofinitary groups under the action of S(kappa).
• To investigate the conditions under which these groups can realize specific partitions of kappa.
• To explore the consistency of universal kappa-maximal cofinitary groups for various cardinalities.

Main Methods:

• Utilizes techniques from combinatorial set theory and descriptive set theory.
• Applies results from higher analogues of Bell's theorem.
• Employs model-theoretic consistency arguments.

Main Results:

• Demonstrates that any kappa-maximal cofinitary group has fewer than kappa orbits under the natural action of S(kappa) on kappa.
• Shows that if p(kappa) = 2^kappa, any partition of kappa into fewer than kappa sets can be realized as orbits.
• Establishes the consistency of a kappa-maximal cofinitary group universal for groups of size less than 2^kappa = lambda, for any regular lambda > kappa.

Conclusions:

• The structure of kappa-maximal cofinitary groups is tightly constrained regarding their orbits.
• The combinatorial properties of kappa, specifically p(kappa), determine the realizability of partitions.
• The existence of universal groups with specific properties is consistent within ZFC set theory.