Year: 2014
Type: article
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 Π20$\Pi ^{0}_{2}$-hard, Σ20$\Sigma ^{0}_{2}$-hard, and lies in Σ30$\Sigma ^{0}_{3}$. We also show that the sets of orders of torsion elements of finitely presented groups are precisely the Σ20$\Sigma ^{0}_{2}$ sets which are closed under taking factors.
Author: Maurice Chiodo
Institution: University of Neuchâtel
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2765057959