On torsion in finitely presented groups

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 ${\mathrm{\Pi }}_2^0$-hard, ${\mathrm{\Sigma }}_2^0$-hard, and lies in ${\mathrm{\Sigma }}_3^0$. We also show that the sets of orders of torsion elements of finitely presented groups are precisely the ${\mathrm{\Sigma }}_2^0$ sets which are closed under taking factors.