Cofinitary Groups and Other Almost Disjoint Families of Reals
Bart Kastermans

TL;DR
This paper explores the existence, structure, and complexity of maximal almost disjoint families, specifically very mad families and cofinitary groups, revealing new results under various set-theoretic assumptions.
Contribution
It establishes existence results for very mad families, analyzes the structure of cofinitary groups, and investigates their complexity and embedding properties under different axioms.
Findings
Existence of many orthogonal families under MA.
Existence of a coanalytic very mad family under V=L.
Maximal cofinitary groups cannot have infinitely many orbits.
Abstract
We study two different types of (maximal) almost disjoint families: very mad families and (maximal) cofinitary groups. For the very mad families we prove the basic existence results. We prove that MA implies there exist many pairwise orthogonal families, and that CH implies that for any very mad family there is one orthogonal to it. Finally we prove that the axiom of constructibility implies that there exists a coanalytic very mad family. Cofinitary groups have a natural action on the natural numbers. We prove that a maximal cofinitary group cannot have infinitely many orbits under this action, but can have any combination of any finite number of finite orbits and any finite (but nonzero) number of infinite orbits. We also prove that there exists a maximal cofinitary group into which each countable group embeds. This gives an example of a maximal cofinitary group that is not a free…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms · Homotopy and Cohomology in Algebraic Topology
