A co-analytic Cohen indestructible maximal cofinitary group

TL;DR

This paper constructs a highly complex maximal cofinitary group that remains indestructible under Cohen forcing, demonstrating its consistency with large continuum sizes and providing a new proof of existing results in constructible universe.

Contribution

It introduces a $ ext{Pi}^1_1$ maximal cofinitary group that is Cohen indestructible, expanding understanding of such groups in set theory.

Findings

Existence of Cohen indestructible maximal cofinitary group under constructibility assumption

Construction of a $ ext{Pi}^1_1$ maximal cofinitary group in $L$

New proof of Kastermans' result using forcing-inspired methods

Abstract

Assuming that every set is constructible, we find a $\Pi^1_1$ maximal cofinitary group of permutations of $\mathbb N$ which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans' result that there exists a $\Pi^1_1$ maximal cofinitary group in $L$.