On torsion in finitely presented groups

Abstract

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

11 pages. This is the version submitted for publication

Cited by in corpus (3)

• A note on torsion length
• Torsion, torsion length and finitely presented groups
• Generalized torsion orders of generalized torsion elements