  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.
