The structure of [Formula: see text] -maximal cofinitary groups

We study [Formula: see text] -maximal cofinitary groups for [Formula: see text] regular uncountable, [Formula: see text] 1. Any [Formula: see text] -maximal cofinitary group has [Formula: see text] many orbits under the natural group action of [Formula: see text] on [Formula: see text] . 2. If [Formula: see text] then any partition of [Formula: see text] into less than [Formula: see text] many sets can be realized as the orbits of a [Formula: see text] -maximal cofinitary group. 3. For any regular [Formula: see text] it is consistent that there is a [Formula: see text] -maximal cofinitary group which is universal for groups of size [Formula: see text] . If we only require the group to be universal for groups of size [Formula: see text] then this follows from [Formula: see text] . .