Cofinitary groups and projective well-orders
Vera Fischer, Lukas Schembecker, David Schrittesser
TL;DR
This work introduces tight cofinitary groups as forcing-indestructible, highly definable witnesses to maximality in the realm of cofinitary groups. It develops a parameter-free orbit-coding framework using Zhang forcing to produce coanalytic tight cofinitary groups, and strengthens this with orbit-coding that works for every new word. The authors prove the existence of tight cofinitary groups under Martin's Axiom and construct models with a $\Delta^1_3$-well-order of the reals and ${\mathfrak{a}}_g$ tightly witnessed by coanalytic groups, while preserving tightness through Sacks-type coding iterations. As applications, they provide a new Miller indestructibility result and demonstrate a uniform method to obtain coanalytic witnesses to ${\mathfrak{a}}_g$ across several cardinal constellations, and discuss several open questions concerning separations and the relationship between different cardinal characteristics associated with cofinitary groups.
Abstract
We introduce the notion of a tight cofinitary group, which captures forcing indestructibility of maximal cofinitary groups for a long list of partial orders, including Cohen, Sacks, Miller, Miller partition forcing and Shelah's poset for diagonalizing maximal ideal. Introducing a new robust coding technique, we establish the relative consistency of $\mathfrak{a}_g=\mathfrak{d}<\mathfrak{c}=\aleph_2$ alongside the existence of a $Δ^1_3$-wellorder of the reals and a co-analytic witness for $\mathfrak{a}_g$.