Definability of maximal cofinitary groups

TL;DR

This paper proves the existence of a highly definable maximal cofinitary group that is isomorphic to a free group, establishing the lowest possible complexity for such groups.

Contribution

It constructs a closed, definably simple maximal cofinitary group that is isomorphic to a free group, clarifying its definitional complexity.

Findings

Existence of a closed, $ ext{Pi}^0_1$ set generating an $F_ ext{sigma}$ maximal cofinitary group.

The maximal cofinitary group is isomorphic to a free group.

This is the lowest possible definitional complexity for such groups.

Abstract

We present a proof of a result, previously announced by the second author, that there is a closed (even $\Pi^0_1$) set generating an $F_\sigma$ (even $\Sigma^0_2$) maximal cofinitary group (short, mcg) which is isomorphic to a free group. In this isomorphism class, this is the lowest possible definitional complexity of an mcg.