In my talk I will give an introduction to (maximal) cofinitary groups (mcg's) and their corresponding cardinal characteristic \(\mathfrak{a}_g\).

I will present a notion of tightness for mcg's which implies the forcing indestructibility for various types of tree forcings, allowing us to prove that \(\mathfrak{a}_g\) stays small in various models.

Further, I will explain Zhang's forcing - the central forcing notion in context of mcg's - and show how one can adapt this forcing by some new coding techniques in order to construct co-analytic tight cofinitary groups. If time permits, we will see how these result may be combined with recent developments regarding projective well-orders and cardinal characteristics to obtain: Consistently, \(\mathfrak{a}_g = \mathfrak{d} < \mathfrak{c} = \aleph_2\) alongside the existence of a \(\Delta^1_3\)-wellorder of the reals and a co-analytic witness for \(\mathfrak{a}_g\).

This is joint work with Vera Fischer and David Schrittesser.