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. (1)

   Any \(\kappa \)-maximal cofinitary group has \({<}\kappa \) many orbits under the natural group action of \(S(\kappa )\) on \(\kappa \).

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