Cofinitary Groups and Other Almost Disjoint Families of Reals

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 group. We start the investigation into which groups have cofinitary actions. The main result there is that it is consistent that $\bigoplus_{\alpha \in \aleph_1} \mathbb{Z}_2$ has a cofinitary action. Concerning the complexity of maximal cofinitary groups we prove that they cannot be $K_\sigma$, but that the axiom of constructibility implies that there exists a coanalytic maximal cofinitary group. We prove that the least cardinality $\mathfrak{a}_g$ of a maximal cofinitary group can consistently be less than the cofinality of the symmetric group. Finally we prove that $\mathfrak{a}_g$ can consistently be bigger than all cardinals in Cicho\'n's diagram.