The Structure of $\kappa$-Maximal Cofinitary Groups

Abstract

We study $\kappa$-maximal cofinitary groups for $\kappa$ regular uncountable, $\kappa ={\kappa }^{<\kappa }$. Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell's theorem, we show that: 1. Any $\kappa$-maximal cofinitary group has $<\kappa$ many orbits under the natural group action of $S(\kappa )$ on $\kappa$. 2. If $\mathfrak{p}(\kappa )=2^{\kappa }$ then any partition of $\kappa$ into less than $\kappa$ many sets can be realized as the orbits of a $\kappa$-maximal cofinitary group. 3. For any regular $\lambda >\kappa$ it is consistent that there is a $\kappa$-maximal cofinitary group which is universal for groups of size $<2^{\kappa }=\lambda$. If we only require the group to be universal for groups of size $\kappa$ then this follows from $\mathfrak{p}(\kappa )=2^{\kappa }$.

12 pages