The Structure of $\kappa$-Maximal Cofinitary Groups

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 $\mathfrak{p}(\kappa) = 2^\kappa$.