Cofinitary Groups and Other Almost Disjoint Families of Reals

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.