17.121 (2010)

Let $G$ be a group whose set of elements is the real numbers and which is “nicely definable” (see below). Does $G$ being $\aleph_\omega$-free imply it being free if “nicely definable” means
$\qquad$ a) being $F_\sigma$?
$\qquad$ b) being Borel?
$\qquad$ c) being analytic?
$\qquad$ d) being projective $L[\mathbb{R}]$?

See https://arxiv.org/pdf/math/0212250.pdf for justification of the restrictions.