On torsion in finitely presented groups

Abstract. We describe an algorithm that, on input of a recursive presentation P of a group, outputs a recursive presentation of a torsion-free quotient of P, isomorphic to P whenever P is itself torsion-free. Using this, we show the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed (first proved by Belegradek). We apply our techniques to show that recognising embeddability of finitely presented groups is Π 2 0 $\Pi ^{0}_{2}$ -hard, Σ 2 0 $\Sigma ^{0}_{2}$ -hard, and lies in Σ 3 0 $\Sigma ^{0}_{3}$ . We also show that the sets of orders of torsion elements of finitely presented groups are precisely the Σ 2 0 $\Sigma ^{0}_{2}$ sets which are closed under taking factors.