  We give a uniform construction that, on input of a recursive presentation $P$
of a group, outputs a recursive presentation of a torsion-free group,
isomorphic to $P$ whenever $P$ is itself torsion-free. We use this to re-obtain
a known result, the existence of a universal finitely presented torsion-free
group; one into which all finitely presented torsion-free groups embed. We
apply our techniques to show that recognising embeddability of finitely
presented groups is $\Pi^{0}_{2}$-hard, $\Sigma^{0}_{2}$-hard, and lies in
$\Sigma^{0}_{3}$. We also show that the sets of orders of torsion elements of
finitely presented groups are precisely the $\Sigma^{0}_{2}$ sets which are
closed under taking factors.
