Tight cofinitary groups

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 ideals. Introducing a new robust coding technique, we establish the relative consistency of a g = d < c = ℵ 2 alongside the existence of a Δ 3 1 -well-order of the reals and a co-analytic witness for a g .